That's a bad mistake by the property manager. Here's the problem: You clearly have a contract (the fact that you pay for parking and trash service makes it contract), and clearly there is a mistake, but now we don't know what the rent actually is - not from the contract, but I assume you agreed verbally on a number with the property manager. If you try to pay no rent, you will not be getting away with it, and things will go to court, and between you and the property manager you will lose out by paying money for lawyers and wasting your time.
If you pay what you think is the agreed rent, and the property manager accepts that, then you should be legally fine - in the unlikely case that your contract went to court, you would point at the the $0 is clearly wrong, and the fact that you paid $X and the landlord didn't complain is strong evidence that the correct amount is $X. In the case that your landlord suddenly realised the contract says $0 (clearly wrong) and he then claims the correct rent was $X + $200 then he would be getting nowhere with this.