There is absolutely nothing that our government does better than private industry- mail delivery, R&D, roads/bridges maintenance, healthcare, etc. Private industry running the parks will be a dramatic improvement (actually won’t take much to improve upon the current state). Let private industry run it like a business for a while and watch and see the improvement.
There are things government does better. I'm not sure private police would do better. The places that are awful I think would be awful with private police also, because the problem is generally voters/residents, and you don't fix that problem by privatizing. I think fire protection would be similar. Probably would do better with private fire protection in non-poor areas because insurance would be a mechanism to align incentives. It's hard to regulate negative externalities privately. Even though most people/companies will do more or less the right thing, it would be harder for them to do so if their weren't consequences for the people/companies who wouldn't.
The parks should be much nicer if they're privatized, but they won't be able to provide park experiences to people of modest means at below cost. Of course, as has been shown, Mississippi's government isn't particularly good at that either. Maybe their can be some programs aimed at providing discounts to those that need it so they can enjoy the parks also.