If possible, you could sign up for an online patient portal with kaiser (idk if they have one, BCBS does) and then sign up for "paperless billing". Usually you can turn those things off and on, just be sure to do it at least 30 days before you expect a bill. this way you can cover your tracks by just by passing their access to the bills going through.
In my experience, all of my bills from my doctor came to my college address that I put down. All of my insurance "health claims" came to my email attached to my online portal. My dad (though fully knowing off all my appointments and bills) never saw a single piece of paper.
Alternatively, you can go see the doctor you're scheduled for and just do a routine consult or blood work. Then just pay attention to where the paperwork gets sent to. Tell your parents you joined some BS club and need physical to get cleared. If it gets sent home you'll have your answer for the big visit. If it doesn't get sent home or show up on a claim then you're good.