In San Francisco at least, food trucks and carts pay the city a fairly hefty fee to operate and park.
It would be difficult to justify a demand system since day passes for parking meters are sold for things like moving trucks well ahead of time (how do you price something on a random Tuesday two weeks ahead of time?). However, the day parking passes are quite expensive (~$200 for 4 spots all day), so I'm not sure that even at peak demand, the city would make more than that in meter fees.
Exactly. Just like concessions at fairs or festivals are handled. Either there needs to be private space to park the trucks or a way to bid for the space. Using public space for commerce free of charge basically means the owner getting an insane discount on some of the most valuable real estate in the world.
When parking is free, drivers are also getting an "insane discount." If drivers can park for free, food trucks should also park for free - but the best solution is that nobody parks for free.
If a food truck can outbid people who want to park, then the city's economy needs food trucks more than parking spots.