It's common for DC to illegally tow properly parked cars and also for them to tow you without recording it at all (thus not being able to tell you where they moved it to you) or telling you the wrong place.

See my past complaints here.

I have since put an AirTag in my car in case they do this again. Worst case scenario, wherever they move it, they often tow you to illegal parking spots, so then they ticket you again, and the new ticket will tell you where the car is.

Dispute the tickets. I have disputed 7 of my 9 tickets and have won all of the disputes, even if the adjudicator gets super snotty about it.