>>233092
RNfag here. I'm not an expert in community health, but I'll tell you what I know.
For anyone over 18, private health information on a patient's medical record can only be released to the patient unless they've signed a waiver that lets others access it. Sensitive material like mental illness and STD screenings are normally considered even more privileged and have their own special sections on the waiver forms.
If you're on your parents' insurance and you file a claim, I believe they will be able to see that something was charged, but not what. If you self-pay (cash or whatever up front), it'll be totally anonymous outside of being recorded in your medical record.
There are also clinics that offer free testing; you can call up your local health department or use Google for the specific policies; it probably varies by state.