It will depend upon state negligence laws if you, the other party or both of you would be found at fault for this accident.

If both drivers in your situation believe the other is at fault and place claims through each other's insurance company then the claims adjusters would take statements, view the damage, and so on to investigate the claim and determine fault.

There is normally contributory negligence or comparative negligence laws in a state.

The concept of comparative negligence is to allow damage recovery reduced by a person’s own percentage of negligence. In some states, such as Ohio, comparative negligence law specifies that if a party is more than 50 percent at fault, recovery is not allowed.

With contributory negligence if you contributed to the accident in any way, like not taking reasonable care according to the insurance company when pulling into the parking spot, then you would be unable to recover from the other party for damages.

You can contact your insurance agent to notify them of the accident and discuss your state's negligence laws and how fault will be determined.