First thing you have to understand is that the other driver's insurance company is not your insurance company and owes you nothing until a court of law says so.
Obviously, the other driver's insurance company can say anything it wants in defense of its insured.
But I disagree that you are at fault.
Generally, when somebody backs up and hits or gets hit, that person is at fault.
Had the parking spaces been at right angles with two way traffic then, yes, you both would have shared fault.
But the way you describe things, backing out on the diagonal where the lane is one way, you can't reasonably expect another car to be coming from the wrong direction while you are looking behind you and to the right.
At this point you have two options:
1 - Sue the other driver in small claims court.
Or
2 - Make the claim on your own collision coverage and let your own insurance company go after the other driver for reimbursement. Keep in mind that your own insurance company also has the right to an opinion that you share fault.

