While Ronny is well meaning the above advise may get the OP in more trouble. Because the car was not stolen it was towed. If they call the police one of several things will happen.
Scenario 1 - They will ask if he knows who stole it. If the OP says NO they will take a report. However, they will also check to see if it was towed. As that is standard procedure. Once they discover it was towed, if they find out that the OP already knew that the police could go after the OP for filing a false report.
Scenario 2 - The OP explains this story to them. They will state that that is a Civil matter between the Tow Company, the Residence Association(if there is one), and the OP. They will not intervene.
Scenario 3 - The OP goes to the lot to pick up the car and they try to make them pay. In this case the cops would come out if they call. However it would be a low priority call. Depending on how busy they are it may be 30 minutes or 6-8 hours, before they show up. Once they show up all the tow company has to say is that the OP has to pay and the cops can't do a thing.
Now, it appears that 3 may not apply as the report is that they want to charge the OP to "bring it back". So should they bring it back if it was a mistake..yes. But if they don't there is nothing the police can do to "force" them too. If the Tow company stated it was a mistake I would hope that they OP gets that in WRITING. Although if the OP ends up paying for them to bring it back and there is no proof that they had to pay for the initial tow that may be enough proof. I am just talking expediency in getting the car back and not who was right. If it was towed by mistake they then need to file a Small Claims case for any losses that was incurred. If the city has an agency or commission that licenses tow companies they need to make a report with them as well.