If you had mice in your home, you could sue, as a landlord owes you a warranty of habitability:
So if you had mice in your apartment, you'd be OK. (The guy who said mice ate his carpet but was billed for it was likely leaving a lot out.)
Does this warranty extend to the parking structure? I don't know. But if I were you, I'd find out, since $1,100 is on the line.
I would not concede that they don't owe you the first 500. This is just bag negotiating. Tell them they owe you the $1,080, attach a receipt and go from there. Starting high will make it easier to end up at $500 than starting at $500 would.
Tell them you need a referral for a landlord tenant lawyer. Then get in touch with the lawyer. Tell him you'll pay him $100 for a half-hour consultation. (He might even answer your question for free if he knows the answer off the top of his head. Despite public perception, lawyers give free legal advice all the time.)
By the way, check out your lease and see what it says about parking. See if it says, "You park at your own risk." Etc. Any lawyer you contact will need to know this.
Good luck and let us know how it works out.