The National Parks (and not all of them are Parks, some are Monuments or Historical Sites and have different status and rules) are each subject to the laws under which they were established -- Some are barred from ever charging admission, for example -- Some set up their own laws and some use the laws of the surrounding state or states (or choose one of them) -- Isle Royale NP, for example, requires a state fishing license
to fish in Lake Superior, but no license
to fish in the lakes and streams on the Island.
The US Gummint owns Yellowstone, but agreed to abide by UN Heritage Site rules with the 1972 treaty -- The UN doesn't own any of the parks and only 'controls' certain aspects of the Sites. The reverse is true, although the UN owns the Headquarters in NYC, it has agreements with the USG about some affairs -- "However, the United Nations is bound by an agreement with its host country to prevent its Headquarters from being used as a refuge for persons who are avoiding arrest under the federal, state or local laws of the United States or who are required by the Government of the United States for extradition to another country or who are trying to avoid the servicing of a legal process." Works both ways.