Many factors determine who will pay for any damage whether you are in an accident in a parking lot or on a public street. Usually the party who is cited for the accident will be responsible for the damages. If the police are not called and the drivers exchange insurance information and the loss is later reported to the insurance company then an adjuster may decide who is liable. It is also possible that the police are not called and the drivers do not exchange insurance information and agree to be responsible for their own damage.
If you were in an accident in a parking lot and feel that the other party was at fault you would make a claim with their insurance company. Their insurer would then investigate and make a determination on her was at fault. If they find that their insured was at-fault than your third party claim for your damages should be accepted and paid out.
If the other party feels that you are at fault they likely will make a claim against your policy and then if your insurer finds you liable they would pay for the repair of damages you caused.
If you came back to the parking lot and found your car was damaged by an unknown car (or even by a cart) then you would have to put the claim through your own Collision coverage so that the repairs, minus your deductible, could be paid for by your insurer.