A boy ran into my stopped car with his bicycle. His parents' home owners insurance paid for the damages.
So I definitely think that this case could go to something like that.
However, I've never known of a police officer to be willing to come to a private parking lot - whether that's a grocery store lot or the lot of my parents' townhouse development - for something like this, making a police report *impossible*. In my case, there was absolutely no problem with this, the homeowners insurance did NOT need a police report.
And I do suppose they need to talk with the owner...thankfully the bicycle-boy's parents were known slightly to my stepdad, and he called to tell them what was going on, and he didn't balk. And when a Suburban driver backed into my parked (and empty) Beetle, I got home and on the phone to insurance before she got home, and they had called and spoken with her unsuspecting husband before she got home as well. But we happened to have the same insurance company so I suppose that made it slightly easier.
While you wait, you might find a good and honest bodywork shop and get some estimates. Just so you're prepared.
In Kentucky they actually have a paper you fill out and file with the state and that is the "police report" for private parking lots. I had to do this with a lady when she backed into my van while I was driving past her vehicle. She pulled into the space and then immediately backed out without ever looking. Of course hearing how many accidents she had previously made me better understand that she did not need a license.