It simply allows you to carry according to the laws of the state wherein the National Park resides, unless the state itself bans carry in the National Park (which none do).
For example, in the Smokies: in NC you can open carry in the park without a permit, or open or conceal carry with a permit. On the TN side, you can do either only with a permit.
Note that however, federal facilities (buildings) within all the parks are off limits for carry nation wide. They are supposed to be posted, but seems there is not a federal standard sign, or exact location for them.