Since it is your parent's policy, they will get a copy of claims paid on that policy.

I don't know if pdoc's billing office would be able to send a bill or account statement only to you, since your parent/the insurance company would be listed in the pdoc's records as being the guarantor, the party responsible for payment. But you could ask, stating you need this to be private.

Your parent won't have any access to the reason you see the pdoc. You could get creative there - maybe say you see the pdoc for a sleep medication or something like that? Then everything would be above board and maybe not so stressful to deal with.