And whose responsibility is it if something goes wrong ?
As a driver I am not going to invalidate my insurance just to save a few dollars in parking. Especially when the person managing this orchestra may not be capable of keeping everything in sync.
Those problems are bureaucratic problems that will be solved with the money at stake. You're already protected with valet parking, through a combination of your own insurance and the valets. This wouldn't be any different, you'd pay a few more bucks per year in insurance to save a large percentage in parking -- it would make economic sense in densely packed cities.
As a driver I am not going to invalidate my insurance just to save a few dollars in parking. Especially when the person managing this orchestra may not be capable of keeping everything in sync.