You normally will put the claim through the owner of the car's insurance policy. Hopefully he or she gave the driver permission to use their vehicle and their insurance policy will accept the claim. The owner has vicarious liability for those that they give consent to use their car.

Usually your insurance would be covered through the owner's property damage liability policy. Insurance follows a vehicle first and driver second so if the claim amount goes beyond the owner's liability limits you can see about using the driver's liability insurance to place a secondary claim.