I've seen 20+ car crashes outside my window in the last few years and cars usually either stay right where they stopped until police (and later tow trucks) arrive in the case of serious accidents or else pull over to the side of the road to exchange insurance information in minor accidents. In some of the moderate cases, I've seen the drivers try to move their cars out of the way of traffic but they still stay at the scene. I've never seen a car pull into a parking lot after an accident despite there being one literally right there where I live.

What country are you in? Is the insurance public, or does where you live suck?

If the parking lots privately owned, or is it owned by the government?

Are you wanting to know only if they need to pay upfront, or are you asking if they will be reimbursed by the insurance?

Is the driver of this car "at fault"?

Depends on your insurance and likely who was at fault.

Technically yes, they must pay. If they are there for a short while like an hour or so nobody would mind, depending on how dickish the owner of the property is.

For more long term parking say a couple days, it's liable to get towed.

I was working as a security guard in a warehouse type building, and there was this one car with a broken windshield that belonged to one of the employees. This car had been parked in the lot for about a month until the building owner had it towed (because he was a prick). Even though parking was free, and it wasn't a restricted area. Legally speaking it was within his right to tow that car, the employee ended up paying about 3-400 bucks to the tow company.