There are two instances prior to this in the books which points to the fact that sometimes Harry does not realize that he is speaking in Parseltongue
First time in "Harry Potter and the Sorcerer's Stone"
The snake jerked its head toward Uncle Vernon and Dudley, then raised its eyes to the ceiling.
It gave Harry a look that said quite plainly:
“I get that all the time.”
“I know,” Harry murmured through the glass, though he wasn’t sure the snake could hear him.
“It must be really annoying.”
The snake nodded vigorously.
Harry Potter and the Sorcerer's Stone, Chapter 2 The Vanishing Glass
Second time in "Harry Potter and the Chamber of Secrets"
“What’s bad?” said Harry, starting to feel quite angry. “What’s wrong with everyone? Listen, if I hadn’t told that snake not to attack Justin —”
“Oh, that’s what you said to it?”
“What d’you mean? You were there — you heard me —”
“I heard you speaking Parseltongue,” said Ron. “Snake language. You could have been saying anything — no wonder Justin panicked, you sounded like you were egging the snake on or something — it was creepy, you know —”
Harry gaped at him.
“I spoke a different language? But — I didn’t realize — how can I speak a language without knowing I can speak it?”
Harry Potter and the Chamber of Secrets, Chapter 11 The Dueling Club
Now coming to the second question.
Finally Harry spoke, causing Hermione to gasp and jump.
“Are you Bathilda?”
I think initially Harry spoke in English. The reason for Hermione's reaction could be because of the fact that they were under the invisibility cloak and yet Bathilda was able to see them and beckoned them. And also when Harry spoke she might have been surprised.
By all the rules of normal magic, meanwhile, she ought not to be able to see Hermione and him at all. Nevertheless, Harry had the strangest feeling that she knew that they were there, and also who they were. Just as he had reached this uneasy conclusion, she raised a gloved hand and beckoned.
Hermione moved closer to him under the Cloak, her arm pressed against his.
Harry Potter and the Deathly Hallows, Chapter 17 Bathilda’s Secret
It becomes clear few lines later
“Are you Bathilda?”
The muffled figure nodded and beckoned again.
Beneath the Cloak Harry and Hermione looked at each other. Harry raised
his eyebrows; Hermione gave a tiny, nervous nod.
Harry Potter and the Deathly Hallows, Chapter 17 Bathilda’s Secret
This implies Hermione understood Harry's question. There are two reasons why I am saying this.
Firstly Harry could speak in Parseltongue when he knew he was talking to a snake.
“Harry,” said Ron. “Say something. Something in Parseltongue.”
“But —” Harry thought hard. The only times he’d ever managed to speak Parseltongue were when he’d been faced with a real snake. He stared hard at the tiny engraving, trying to imagine it was real.
“Open up,” he said.
Harry Potter and the Chamber of Secrets, Chapter 16 The Chamber of Secrets
Secondly we know Voldemort speaks to Nagini in english and she understands it but cannot reply in English.
“Dinner, Nagini,” said Voldemort softly, and the great snake swayed and slithered from his shoulders onto the polished wood.
Harry Potter and the Deathly Hallows, Chapter 1 The Dark Lord Ascending
Back to the question
“Come!” called Bathilda from the next room.
Hermione jumped and clutched Harry’s arm.
Harry Potter and the Deathly Hallows, Chapter 17 Bathilda’s Secret
This part of the conversation would have happened in Parseltongue because Bathilda (Nagini) cannot speak English and she did not want Hermione around. And as @albusseverus potter puts it, Hermione might have heard it as "muffled mumbling of an extremely old lady". She did not realize that the language for Parseltongue because few lines later
“Who is this man?” he repeated loudly.
“Harry, what are you doing?” asked Hermione.
“This picture, Hermione, it’s the thief, the thief who stole from Gregorovitch!
Please!” he said to Bathilda. ”Who is this?”
“Why did you ask us to come with you, Mrs.—Miss—Bagshot?” asked Hermione,
raising her own voice. ”Was there something you wanted to tell us?”
Harry Potter and the Deathly Hallows, Chapter 17 Bathilda’s Secret
Conclusion:
Harry spoke in English till he was alone in the room with Bathilda (Nagini). It was the situation which made Hermione react the way she did. So in short Hermione did not realize that they were speaking in Parseltongue because Bathilda (Nagini) hardly spoke in Parseltongue when Hermione was around