Every time I use my insurance, I get both a physical and an electronic explanation of benefits. The electronic one usually arrives a few days earlier than the mailed copy. It lists what doctor I went to and what general code the service falls under, as well as what insurance paid and what my expected contribution to the expenses is. My insurance also has charges on the landing page for my policy that shows how much they and I have spent towards my medical expenses for the year so far. A few weeks after that, I usually get a bill in the mail for lab work and anything beyond my copay due to the doctor's office.
So if your parents is set up the same way, you'd have a very hard time hiding it from them. You would need to intercept their mail, delete their email, and hope that they never have a need to login to their health insurance website. If you are a minor, you would also likely need one of them to sign off for their consent for you to receive treatment. It is most likely not going to be possible to use insurance without the policyholder knowing about your visit.
What does the $650 include? Is this a visit with the doctor and lab work? Because that is sadly, probably about right in terms of what those things cost without insurance. You could ask if they do a sliding scale option for low income patients.
If you're young and not established in a career yet, I'd consider looking into jobs at places with inclusive coverage like Starbucks, Target, Best Buy, etc. Those are all nationwide chains with very low requirements for entry that would get you coverage for everything including surgery, so you wouldn't be reliant on your parents.
