That's due to some mysterious decisions on part of the Show-team.
In the books it didn't quite happen that way and because the Show chose to skip the whole episode about:
Tysha
So naturally, Tyrion's reaction doesn't make sense to a show-fan. Furthermore, he did take a dagger and a crossbow from his father's chambers.
Let's see how things happened in the books. Spoilers beyond this point.
Tyrion's Wife
When Tyrion was a teenage boy, he and Jaime were riding from Casterly Rock when they happened upon a few rapists chasing a young girl. The rapists ran away when they saw Jaime but Jaime was not satisfied. It offended him that ruffians would prey upon innocent travellers this close to Casterly Rock. So he went back to get help and left the girl in Tyrion's care.
Tyrion took her to an inn and found out that the girl was named Tysha and was a crofter's daughter who was on her way to nowhere really as her father had died and she had no immediate family. Tyrion fed her, comforted her and before they knew it, they were sharing a bed. Tyrion recalls that she was very lovely. She was a virgin and by morning, Tyrion was in love with her.
Tyrion married her by bribing a drunk septon. But he was very well aware that his father would never accept a Crofter's daughter as his daughter-in-law, wed to second in line to the throne of the ancient Kings of the Rock. So he found a way but it didn't work out very well. He set her up in a cottage and they played on being husband and wife for few weeks until the Septon he had bribed to marry them told Lord Tywin.
Then Tywin crushed him. He made Jaime confess that he had hired a virgin whore to make a man out of Tyrion. Then he gave her up to his garrison and bid Tyrion watch. Then he had Tyrion take her as well.
Tyrion hated his father for that. But it all made sense, right? He only had to look at himself. Why would a girl like Tysha want to be with a hideous malformed dwarf like him?
For complete account, read AGOT Tyrion VI.
Jaime's Confession
Before setting Tyrion free, Jaime confessed that Tysha was not a whore, she was just a lowborn girl they met. Tyrion was now consumed by rage and guilt, his own family had deceived him and he himself took part in raping his wife and one true love.
"Tysha?" His stomach tightened. "What of her?"
"She was no whore. I never bought her for you. That was a lie that
Father commanded me to tell. Tysha was . . . she was what she seemed
to be. A crofter's daughter, chance met on the road."
Tyrion could hear the faint sound of his own breath whistling hollowly
through the scar of his nose. Jaime could not meet his eyes. Tysha. He
tried to remember what she had looked like. A girl, she was only a
girl, no older than Sansa. "My wife," he croaked. "She wed me."
"For your gold, Father said. She was lowborn, you were a Lannister of
Casterly Rock. All she wanted was the gold, which made her no
different from a whore, so . . . so it would not be a lie, not truly,
and . . . he said that you required a sharp lesson. That you would
learn from it, and thank me later . . ."
"Thank you?" Tyrion's voice was choked. "He gave her to his guards. A
barracks full of guards. He made me . . . watch." Aye, and more than
watch. I took her too . . . my wife . . .
"I never knew he would do that. You must believe me."
"Oh, must I?" Tyrion snarled. "Why should I believe you about
anything, ever? She was my wife!"
“Tyrion –it...”
He hit him. It was a slap, backhanded, but he put all his strength
into it, all his fear, all his rage, all his pain. Jaime was
squatting, unbalanced. The blow sent him tumbling backward to the
floor. “I... I suppose I earned that.”
"Oh, you've earned more than that, Jaime. You and my sweet sister and our loving father, yes, I can't begin to tell you what you've earned. But you'll have it, that I swear to you. A Lannister always pays his debts." Tyrion waddled away, almost stumbling over the turnkey again in his haste. Before he had gone a dozen yards, he bumped up against an iron gate that closed the passage. Oh, gods. It was all he could do not to scream.
ASOS - Tyrion IX
He lied about killing Joffrey to hurt Jaime. He also told him about Cersei infidelity and then went on his way.
Jaime handed him the ring of keys. “I gave you the truth. You owe me
the same. Did you do it? Did you kill him?”
The question was another knife, twisting in his guts. “Are you sure
you want to know?” asked Tyrion. “Joffrey would have been a worse king
than Aerys ever was. He stole his father’s dagger and gave it to a
footpad to slit the throat of Brandon Stark, did you know that?”
“I... I thought he might have.”
“Well, a son takes after his father. Joff would have killed me as
well, once he came into his power. For the crime of being short and
ugly, of which I am so conspicuously guilty.”
“You have not answered my question.”
“You poor stupid blind crippled fool. Must I spell every little thing
out for you? Very well. Cersei is a lying whore, she’s been fucking
Lancel and Osmund Kettleblack and probably Moon Boy for all I know.
And I am the monster they all say I am. Yes, I killed your vile son.”
He made himself grin. It must have been a hideous sight to see, there
in the torchlit gloom.
ASOS - Tyrion IX
He met Varys in the tunnels. That's when he realized that Shae talked about these tunnels which opened directly into chambers of the Hand, given how Varys used to smuggle her inside Maegor's holdfast.
He told Varys he cared naught for his life or anyone else's at this point. Despite Varys' warnings, Tyrion made him tell him the secret entrance to his father's chambers.
The juncture was otherwise empty, but on the floor was a mosaic of a
three-headed dragon wrought in red and black tiles. Something niggled
at Tyrion for a moment. Then it came to him. This is the place Shae
told me of, when Varys first led her to my bed. “We are below the
Tower of the Hand.”
“Yes.” Frozen hinges screamed in protest as Varys pulled open a long
closed door. Flakes of rust drifted to the floor. “This will take us
out to the river.” Tyrion walked slowly to the ladder, ran his hand
across the lowest rung. “This will take me up to my bedchamber.”
“Your lord father’s bedchamber now.”
He looked up the shaft. “How far must I climb?”
“My lord, you are too weak for such follies, and there is besides no
time. We must go.”
“I have business above. How far?”
“Two hundred and thirty rungs, but whatever you intend-”
“Two hundred and thirty rungs, and then?”
“The tunnel to the left, but hear me -”
“How far along to the bedchamber?” Tyrion lifted a foot to the lowest
rung of the ladder. “No more than sixty feet. Keep one hand on the
wall as you go. You will feel the doors. The bedchamber is the third.”
He sighed. “This is folly, my lord. Your brother has given you your
life back. Would you cast it away, and mine with it?”
“Varys, the only thing I value less than my life just now is
yours. Wait for me here.” He turned his back on the eunuch and began
to climb, counting silently as he went.
ASOS - Tyrion IX
There he found Shae in bed of Tywin Lannister, the man who publicly despised Whores and Whoremongers. Shae had sold out to Cersei to see Tyrion hanged and had the bright idea of calling him "My Giant of Lannister" at that time, and died for it.
“M’lord?” a woman’s voice called.
That might have hurt me once, when I still felt pain. The first step
was the hardest. When he reached the bed Tyrion pulled the draperies
aside and there she was, turning toward him with a sleepy smile on her
lips. It died when she saw him. She pulled the blankets up to her
chin, as if that would protect her. “Were you expecting someone
taller, sweetling?”
Big wet tears filled her eyes. “I never meant those things I said, the
queen made me. Please. Your father frightens me so.” She sat up,
letting the blanket slide down to her lap. Beneath it she was naked,
but for the chain about her throat. A chain of linked golden hands,
each holding the next.
“My lady Shae,” Tyrion said softly. “All the time I sat in the black
cell waiting to die, I kept remembering how beautiful you were. in
silk or roughspun or nothing at all...”
“M’lord will be back soon. You should go, or... did you come to take
me away?”
“Did you ever like it?” He cupped her cheek, remembering all the times
he had done this before. All the times he’d slid his hands around her
waist, squeezed her small firm breasts, stroked her short dark hair,
touched her lips, her cheeks, her ears. All the times he had opened
her with a finger to probe her secret sweetness and make her moan.
“Did you ever like my touch?”
“More than anything,” she said, “my giant of Lannister.”
That was the worst thing you could have said, sweetling.
ASOS - Tyrion IX
Then he picked up weapons:
Afterward he found Lord Tywin’s dagger on the bedside table and
shoved it through his belt. A lion-headed mace, a poleaxe, and a
crossbow had been hung on the walls. The poleaxe would be clumsy to
wield inside a castle, and the mace was too high to reach, but a large
wood-and-iron chest had been placed against the wall directly under
the crossbow. He climbed up, pulled down the bow and a leather
quiver packed with quarrels, jammed a foot into the stirrup, and
pushed down until the bowstring cocked. Then he slipped a bolt into
the notch.
ASOS - Tyrion IX
Then Tyrion found his father and asked him where Tysha went. Tywin called her a whore. Tyrion warned him against using that word. Tywin did it again. Tyrion shot him and then left.
“We can talk here just as well. Perhaps I don’t choose to go to the
Wall, Father. It’s bloody cold up there, and I believe I’ve had enough
coldness from you. So just tell me something, and I’ll be on my way.
One simple question, you owe me that much.”
“I owe you nothing.”
“You’ve given me less than that, all my life, but you’ll give me this.
What did you do with Tysha?”
“Tysha?”
He does not even remember her name. “The girl I married.”
“Oh, yes. Your first whore.”
Tyrion took aim at his father’s chest. “The next time you say that
word, I’ll kill you.”
“You do not have the courage.”
“Shall we find out? It’s a short word, and it seems to come so easily
to your lips.” Tyrion gestured impatiently with the bow. “Tysha. What
did you do with her, after my little lesson?”
“I don’t recall.”
“Try harder. Did you have her killed?”
His father pursed his lips. “There was no reason for that, she’d
learned her place... and had been well paid for her day’s work, I seem
to recall. I suppose the steward sent her on her way. I never
thought to inquire.”
“On her way where?”
“Wherever whores go.”
Tyrion’s finger clenched. The crossbow whanged just as Lord Tywin
started to rise. The bolt slammed into him above the groin and he sat
back down with a grunt.
ASOS - Tyrion IX
So now you understand, why did Tyrion do it. It was emotional trauma, anger and guilt which lead him to do what he did. People tend to act recklessly in such circumstances.
