Yes, it matters who was at fault in the parking lot accident. If the other party was at fault then you would normally want to make a claim through their property damage liability coverage.

If you had collision coverage then you could make a claim that way and then let your insurance company subrogate against the at fault party or their insurer. Since the other driver has collision coverage this means that party could place their damages through their own insurance company.

You cannot place a claim through your own insurance carrier since you do not have collision coverage. Thus you would need to put it through the other person's insurance since they were the at fault party in the accident. If for some reason they were not found at fault by their insurance provider and your claim was denied then you would either be left to pay for the repairs yourself or take the at fault party to court and try to get a judgment against them for the cost of your repairs.