I assume these are public parking areas for commuters, so the quick answer is NO, but you would need to find out from the local municipality that owns the lot if they have ordinances about work of any type can be done on public land. If you are a legit detailing service (i.e., you are a registered LLC , company, enterprise; you file quarterly taxes with local, state, and federal government; AND you have business insurance), I would also check with your business insurance as they may not cover liability of your work done on publicly owned land OR you may need a rider policy to allow you to do this IF the municipality allows it.
I say this because in a parking lot USUALLY some other vehicle is parked next to the client`s vehicle you are likely to work on. I assume that the client has not given you a key to his vehicle, but that is another liability for both you (and your insurer) and the client to discuss, so moving a vehicle to a remote part of the lot may not be possible. or there may be no room to do so.
If you are doing this as side job for cash, I HIGHLY suggest you refrain from taking on mobile detailing jobs ANYWHERE for clients who are friends-of-friends or referred to you from your clients you may have done work for. Without insurance, one bad detailing "accident" (like a deep scratch in the paint or ripped leather seat from the watch or ring you forgot to take off!!) can cause you to loose a lot of money, since you are assuming that liability without business insurance.
Also, are you taking pictures of the vehicle BEFORE you start detailing it??? If you do not inspect the vehicle closely and photo (or video)-document the condition of the vehicle outside and inside with all of its scratches and dents or hanging trim or missing emblems or ripped upholstery, the owner might blame you as the culprit. You really need to cover your tail when detailing for "other" people (i.e., vehicle owners you do not know).
