If you are backing out of a parking spot and another person cuts in behind you to get into a parking space who is at fault if you back into them?

If you are both moving, the one At Fault is whoever the police officer says is at fault on the accident report. That's all I can tell you with this information. I hope you called the police. They will say you backed into them and you will say they ran into you. Without a police report faulting one of you, the insurance company will say both are 50% at fault which means each pays for their own damage.