As per the comment, this may depend on where you work. That said, you absolutely have the right to say to your boss "I want parental leave - I'd rather not discuss this with my co-workers. Is it all right if we describe it as annual leave?"
Personally, I think it's fine to make that request - but accepting it is a different matter. Suppose one of your co-workers comes up to your boss and says
- Why is Dave allowed so much time off? I've been here longer, I should get more holiday!
- Has Dave been placed on administrative leave? What did he do?!
- Is Dave ill? Should we get a card and flowers?
- Was Dave fired? Should we start looking for a replacement?
- I heard a rumour that Dave's on study leave - how do I apply for that?
- Where's Dave?
Ultimately, your personal life is your own business. As a manager, I would certainly respect your privacy - but it could become complex without a suitable "cover story."
If I were you, I'd accept the fact that your co-workers may wish to buy you a cake and give you a "congratulations" card.