If it is possible to get law enforcement to respond to the parking lot to take an accident report for a hit and run it is helpful for your insurance carrier but normally it is not mandated by an insurance company.

Some law enforcement jurisdictions do not come out or take police reports for parking lot accidents so insurance companies usually will take a claim without the report. The claim should be made under your physical damage coverages. If it was definitely a car that struck your vehicle is would be your collision coverage. If instead the damages were done by something else other than a car it may fall under your comprehensive coverage.

Whether your rates will go up do to a not at-fault claim will depend upon state insurance laws as well as your insurance company's guidelines. Your state insurance regulatory body can regulate what can and cannot affect insurance rates, beyond those guidelines it is up to the insurance company's rating system.

Typically if you are deemed not at fault you are not surcharged, meaning that you do not have insurance points placed toward your insurance rate. But it may be possible to lose a no claims discount or be pushed into a different rating factor by your insurer due to the claim which later could cause your premium rate to be affected. To find out about your insurance carrier's rating system ask your insurance agent for details.