I've not heard of a clinic that asks parents to go pick up the results before.
But what Lisa said is correct, and here's why: so long as you are a minor in your parents care, THEY are responsible for your health. That's not something I think is unreasonable, in fact, it very much protects you. If you got hit by a bus today, I really doubt you could pay tens of thousands of dollars in hospital fees. In addition, most children and teens (especially chilrden) couldn't and wouldn't initiate their own health care, nor handle all the follow-up.
My feeling is this: if you're really ready to handle the whole of your life, it shouldn't be on someone else's tab. You can (at 16 in this country) go through the courts and apply for emancipated minor status, which, if you won or your parents agreed, would tender you a legal adult in matters like these, and you could go get an apartment, get your own doctor, pay your own bills, and be responsible for yourself. But if someone else is still feeding you, putting the clothes on your back, the roof over your head, and taking care of you when you mess up, basically, it isn't unreasonable to give them the rights and tools TO provide that care.