When he arrives and parks in a tow-away zone, a nearby officer threatens to seize the suspicious-looking vehicle. Parker wasn't planning to keep it, so he shrugs it off. Peter just gave that officer some extra motivation to run the car's plate number and, were that to happen, the cop would discover that it was literally just involved in a high-speed chase with shots fired in downtown Manhattan, including the destruction of multiple police cars.
Even the Blues Brothers think they need to tone it down.
Then Peter would have two options: lie about driving the car during the chase and plead guilt to a litany of crimes, or admit to being Spider-Man and plead guilty of a litany of crimes. Saving five minutes risked Peter's anonymity and the safety of his loved ones, plus hugely inconvenienced the very police force he's trying to help. But then again, it was a pretty sweet car. Who are we to judge?