Bush II actually did a lot of that already. Much of the fee yielding aspects of the big parks have already been sold/leased. I'm not sure any of the lodges or concessions are still run by the park service, and at the larger parks the campgrounds/etc are also run by various private companies taking their slice.

Even the reservation system as we were recently discussing now has a corp entity eating the majority of the "profit".

And the NPS tries to put a positive spin on the whole thing, cause we all know big corps are more efficient (/snark) https://www.nps.gov/orgs/csp/index.htm