Who has right of way in a parking lot? My insurance company claims the person backing up always has right of way.

The insurance company has it backward (pun intended). Right-of-way is with the vehicle in the road. Seems as if they are trying to strong-arm you. If your damage is minimal, simply file suit in small claims court. Good luck.

Actually, it depends on where you were in relation to the car backing up. Typically, it's the car that is backing up which would be responsible for hitting another vehicle that is travelling properly in the marked or unmarked lanes of a parking lot. There are no hard and fast rules that apply, other than the general rules pertaining to keeping a proper lookout (to the rear) and backing up safely. Because it happened in a parking lot (private property), there is usually no police report to "objectively" assess blame/responsibility. The other driver admitted he didn't see you so he should get the blame for failure to keep a proper lookout. The insurance company is blaming you because that is what some insurance adjusters and companies will do, even if it is not supported by the evidence.

Generally, the car backing should keep a proper look-out for other cars. I agree that insurance companies will make ridiculous arguments just to pay less on a claim. I've even seen cases where their insured rear-ends a car and tries to blame the car that was rear-ended.

I am not aware of any rule of the road or statute that gives someone the right-of-way while backing out of a parking space. Technically, a parking lot is not a public roadway, so a police officer should not be issuing tickets to any party in such a situation. However, the person traveling on a public roadway will have the right of way in relation to someone pulling out in front of them. Likewise, even though it's a parking lot, absent a stop sign or other traffic control device, someone who is established in a lane of travel will have the right-of-way over someone pulling out of a parking place.