If you hit a parked car do you file a claim or does the owner file the claim on your insurance?

In most cases, either of you can make the claim. Liability is clear here, so most likely your carrier will take the information from the owner of the parked car and go from there. Your carrier would still expect you to cooperate with them as far as verifying the facts-of-loss, etc.

There are a few carriers, however, who will only accept claims from the named insured. I'm not sure why, as this puts their insured at risk for being sued by the "injured" party, but most of them will take the information any way they can.