It seems to be implied that Tom Riddle actually controlled the Basilisk (ordered it what to do). Same with Voldemort and Nagini.
In the episode in CoS when Harry was telling the snake to leave his duel opponent alone, was the snake obeying Harry's orders, or merely listening to him?
In other words, could Harry order the snakes around like Voldemort, or merely had ability to speak to them and the snake had a choice to listen or refuse?
I think so, yes. If we revisit the scene at the duelling club, from Harry's point-of-view, the snake obeys him implicitly.
Harry wasn't sure what made him do it. He wasn't even aware of deciding to do it. All he knew was that his legs were carrying him forward as though he was on castors and that he had shouted stupidly at the snake, 'Leave him!' And miraculously - inexplicably - the snake slumped to the floor, docile as a thick black garden hose, its eyes now on Harry. Harry felt the fear drain out of him. He knew the snake wouldn't attack anyone now, though how he knew it, he couldn't have explained.
Harry Potter and the Chamber of Secrets - p.145 - Bloomsbury - Chapter 11, The Duelling Club
Now you will remember as well as I, that other people, including Justin Finch-Fletchley himself, didn't see things in this way.
'All I saw,' said Ernie stubbornly, though he was trembling as he spoke, 'was you speaking Parseltongue and chasing the snake towards Justin.'
Harry Potter and the Chamber of Secrets - p.149 - Bloomsbury - Chapter 11, The Duelling Club
But I genuinely think that's born of their fear and their horror at Harry being a Parselmouth. Certainly, one moment the snake's about to strike:
Enraged, hissing furiously, it slithered straight towards Justin Finch-Fletchley and raised itself again, fangs exposed, poised to strike.
Harry Potter and the Chamber of Secrets - p.145 - Bloomsbury - Chapter 11, The Duelling Club
And the next it's just lying there. Justin even has time to have a bit of a yell, instead of getting viciously bitten by a snake:
He looked up at Justin, grinning, expecting to see Justin looking relieved, or puzzled, or even grateful - but certainly not angry and scared.
'What do you think you're playing at?' he shouted, and before Harry could say anything, Justin had turned and stormed out of the Hall.
Harry Potter and the Chamber of Secrets - pp.145-6 - Bloomsbury - Chapter 11, The Duelling Club
Of course, all humans, if you speak their language, can choose to do as you say, or not do as you say. Maybe the same holds for snakes. I don't think this scene proves anything conclusively, but this snake was enraged and about to strike. Maybe Harry's more imperious in Parseltongue than he comes across, but he certainly doesn't sound like he gave out a booming, commanding order. It says that, "[not] even aware of deciding to do it," he "shouted stupidly at the snake". I think the 'stupidly' here means foolishly, as in, he thought it was dumb to go yelling at a snake that's about to attack, but the point is, he doesn't seem to have had a lot of confidence in his actions, so I doubt he came across as very persuasive.
But, y'know, one incident doesn't prove a generalised ability to control snakes and I would imagine that they wouldn't all listen to him all the time. One quote that I find very suggestive, though, is from Dumbledore, talking about Nagini and Voldemort's relationship with her:
I think he is perhaps as fond of her as he can be of anything; he certainly likes to keep her close and he seems to have an unusual amount of control over her, even for a Parselmouth.'
Harry Potter and the Half-Blood Prince - p.473 - Bloomsbury - Chapter 23, Horcruxes
As for the Basilisk, I think that's a special case.
Hermione's hand was back in the air.
'Sir - what exactly do you mean by the 'horror within' the Chamber?'
'That is believed to be some sort monster, which the heir of Slytherin alone can control,' said Professor Binns in his dry, reedy voice.
Harry Potter and the Chamber of Secrets - p.115 - Bloomsbury - Chapter 9, The Writing on the Wall
Now it's possible that the legend says that only the heir of Slytherin could control it because the heir of Slytherin will be a Parselmouth, but nobody else is likely to be. So it could be Parseltongue that gives the heir of Slytherin control. Indeed, the original Fantastic Beasts & Where to Find Them says:
[S]ince Basilisks are uncontrollable except by Parselmouths, they are as dangerous to most Dark wizards as to anybody else, and there have been no recorded sightings of Basilisks in Britain for at least four hundred years.
Fantastic Beasts & Where to Find Them - p.4 - Bloomsbury
This adds further weight to the idea that Parselmouths (and therefore Harry) can control snakes.
As for whether Harry Potter could have controlled Slytherin's monster, it's unclear. As others have stated, it seems like the heir of Slytherin would have a bit more authority, and trying to command or reason with the snake in the Chamber may have been pretty fruitless.
It may also be that Slytherin's monster will only answer to you if you're Slytherin's heir - irrespective of whether or not you're a Parselmouth - although clearly this would be peculiar to that Basilisk, not Basilisks generally.
Personally, I doubt Harry would have been able to call the beast off and I see no reason to be surprised that it never occurred to him to try.
As for Nagini, she is Voldemort's pet and, latterly, a Horcrux. She's a very special case indeed, and I can't see her listening to just any Parselmouth.
Snakes listen to Harry, such as the instance with Justin Finch-Fletchly in CoS. If they do as he commands, that is of their own accord, not because of force.
"Listen, if I hadn't told that snake not to attack Justin..." - (Chamber of Secrets)
This quote sounds as if Harry understood the snake to be actually obeying what he told him to do.
The basilisk is different. Tom Riddle specifically mentions that it only listens to him because he is the heir of Salazar Slytherin.
Basically, Harry has the ability to communicate with snakes. He can speak their language and understand what they say.
This doesn’t give him ‘control’ over them otherwise he could have commanded Nagini to ‘hold’ Voldemort until all the Horcruxes had been destroyed thus making his life easier (albeit making the story a little bit less exciting).
