One reason that many cities require permits for "pros" to shoot in public parks, is because the major production of model, lights, assistants, photographer, etc, is something that interferes with the normal flow of park activities and adds extra risks of law suits for the city, and so forth. So the fees help offset this extra risk.
Generally the wedding shoots I've seen in parks, or the ones I've done myself, have been very "low key" in that they didn't use an assistant, a tripod, etc. There would often be a flash on a bracket and maybe one reflector (often held by the subjects). Even then, you'll occasionally get some "official" who will ask about licenses. You then, just move on.
On the other hand, imagine shooting these shots of Haley in a park (even with a permit). Can you say crowd control problems?
Re: Shooting Permits
Cheers,
rfs