I think that there should be some clarification made here. Are you asking if the Feds should hand over all aspects of a park and say "Here, have at it?" Or are you asking if certain specific operations of a park should go private?
Private companies are in business for one reason only. Profit. With the size and responsibility of running a National Park, no company could completely run it and make a profit without making access and use fees so high that no one will use the park.
On the other hand, if a private company can offer garbage clean-up services or concession services, for example, at less cost than the service is currently operating at, then great. Let them get the contract. But for this to work, the Park Service must maintain management of the park and grow the balls (or have legislation in place) to hold the private company accountable for their work. There must be severe penalties imposed upon the company if they do not live up to the quality of work outlined in the contract.
All too often the government will award a job to the lowest bidding private company, and then do nothing about it when they fail to live up to their end of the deal. Just look at our recent stadiums built in Seattle. The taxpayers are paying the price for unreasonably low construction bids, and the state did nothing about it. When companies bid the jobs, make them stick to the bids, or sue the h*ll out of them like we do in the private sector.
Just my opinion.
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If we are not supposed to eat animals, then why did God make them out of meat??