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It doesn't have to be! Parking enforcement wasn't always within the NYPD's remit. Until 1996, NYCDOT was responsible for parking enforcement.
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Sadly, even if this were the case, there is no mechanism for one NYC city agency to fine another one.

This is why the sidewalks next to city parks take forever to get shoveled after a big snow storm.

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I would settle for a bit of towing, or even just fining the individual cops who park their personal cars on the sidewalk. Parking of city-owned vehicles is also a problem, but the former seems easier to enforce per your point.
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Not to mention completely opaque tinted windshields and windows, and intentionally defaced or corroded license plates..
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Corruption is part of the job description, for the NYPD.
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