Consider a band that wants to hold a concert at that park. They would need a permit
This is nothing like that. If a band want to hold a concert in the park and sell tickets then they want to be able to exclude people from an area of the park, it's obvious that you shouldn't be able to do that without a permit. Even if it's a free concert then you at least want to be able to exclude people from your stage. Also, you're going to be making a lot of noise which is going to affect everyone around you.
There's none of that with Pokemon Go or similar. The only effect you have is that a bunch of people go to the park but that's pretty much the point of public parks.
This looks to me like a case of "Go for the deep pockets". Some guy dropped some litter on your park. You could fine him for it but that requires some work. Maybe he was playing an AR game, or maybe he was using an exercise app which requires a bit of open space. Lets go for those companies because they are rich, even though they had absolutely nothing to do with the litter dropping.