There are two purposes I can see for a permit system:
1. It provides a framework to limit activities in the park, and
2. it can provide a source of funding for the park.
I think in the case of the former, there are some activities one does not want done: such as building a structure. Other activities such as littering does not require the use of a permit, as normal laws quash that.
In the case of the latter, it is entirely possible to just have an entrance fee for the park – if that is legal.
Ultimately, I don't see a need for either need, as unsuitable excess will either happen rarely or can be punished/prevented through normal laws.
Based on the text of the story, it seems to me that the park staff have no reason other than they don't like it. Which is not a valid reason in this context.