Help Please 😬

I'm 21, living on my own, and have a boyfriend that I'm sexually active with. He was my first, so this is all a little new to me. I want to start taking birth control, but I'm still on my parent's insurance. They're super strict and even though I'm an adult, they would kill me if they found out I was having sex. My question is, if I ask my doctor to put me on birth control, will that show up on my parent's insurance? Is it really expensive to buy without insurance?