National Park service, National Forest service, Bureau of Land Management and quite a few other Federal agencies follow state laws when it comes to weapons on their lands. If the state does not allow it, the federal agency will not allow it.
Yes, Federal laws usually trump state laws, but in this case, Federal laws just follow along with state laws.
Maine is currently trying to outlaw weapons in the National park there, and the park service has stated, they will abide by the law if passed. About eight or ten years ago, NC did not allow carry into a National Forest in their state, and the National Forests would not allow concealed carry due to state law.
Some examples, in Utah and Colorado, you can carry open or concealed in National Forests, National parks and BLM land. However, if a state does not allow open carry such as in TX, you cannot carry open in a National Park of Forest in TX (discounting hunting seasons); however, you can carry concealed.