I have to say I have trouble with the they're operating a business and should therefore pay theory. What about a nanny who takes her children to a park. Does they need to pay a fee? What about a business meeting taking place on a park bench?
How about if a running club uses a sidewalk or pathway? Do they need to pay to use these services, because they are operating a business? Should a delivery truck have to pay to use public roads, and only personal vehicles, on personal trips, be allowed to use public roads free of charge? (other than taxes of course!
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In the case of a boot camp, their use of the facility requires no extra effort or administraion on the part of the City, and the revenue would be so inconsequencial, it doesn't seem worth it. Not to mention, I don't think we want to have "roving patrols" as one poster suggested, hasseling everybody in the park to see if they have and/or require a permit to use a park space.
A others have mentioned, booking a space is one thing (wedding photos), or if you require security or fencing (festival or concert), or you require extra administration (food vendors). In my mind, this is simply a cash grab.