When parking in a car park in a city centre there should be another ticket machine avaliable. You would purchase a ticket for a few pounds (dollars etc...) and there would be a key pad so you could print the time you expect to leave. You would display this ticket in your car and go to work or go shopping.

Now in this time where you are busy and your car is just sat there a person would come round, see that you have paid for your car to be hand washed before the time stated on the ticket.

So, no time has been wasted as you were shopping/ working while some nice person was washing your car.

You could extend it to different levels of washing, to include a wax, etc... and there would need to be signs showing the length of time you're required to be away from your vehicle to ensure it is washed satisfactory.

It could work!