Well since he'll most likely want you to see a therapist or counselor of some sort and your parents will most likely have to pay for it; I'm assuming that's a definite possibility.
However, he might just say "I think [your name] should talk to someone just to sort out his mind a bit.] Because seeing a counselor doesn't automatically mean you're depressed, they can't know.
I mean, you can tell your doctor you think you're depressed but he's not going to be diagnosing it. So why should he tell your parents?
Now, with that being said. It's unlikely that your parents wont know that you're seeing someone so they might eventually get caught up and find out you're depressed on their own. Who knows, they may already know. Either way, i don't think you should be so terrified if they do know. If they're good parents you might even want them to know. That way you get double the support. I wish i had had that when i was your age.
Anyway, you shouldn't let the possibility stop you from telling your doctor because no matter what you'll need the help and since your asking I'm assuming you want it.