Who is at fault when you just start to back out of a parking space and a car trys to pull in to the space next to you hitting your passanger side rear bumper?

hard to say without seeing it. technically they have the right of way in the lane and you are responsible for backing out safely. if they just spun through the lot and turned into the space after you had started out, then you have a case. did your bumper hit the side of their car? they were likely 1st then.