You must pay cash and leave your insurance card at home. (IF) you use the insurance, then the insurance sends out a EOB that would state that (you) went to planned parenthood or a clinic on such and such date. Now it may only show for instance a charge for an exam, or that they took a lab test that they billed to the insurance. But either way, you don't want them to ask you, why did you go to the clinic or planned parenthood on such and such date?
But, (if) they have to take a test to determine if in fact you got something like a STD/STI, then the lab (might) mail the results back to you and if so, then I suggest you give them an address of a friend to get it vs getting mailed at your house, since you live at home. Even if your parents don't read your mail and they see something from a lab addressed to you, then could question (why)?. I say this, since when I was a teen went to planned parenthood for birth control, but since I needed a pap, the lab sent back the results to my home, thus when my mom saw the lab addressed to me, asked me why?.