Marco, a couple things:

State laws do apply on Federal lands. The Park Service (as well as the Forest Service, BLM, Fish and Wildlife Service, etc.) all have regulations that apply to their lands (such as the one being discussed) but state laws still apply. Think of license plates and driver's licenses. The Feds do not issue license plates if you live in a Park, and they honor all other states' plates and licenses...

Some places it is ridiculously easy to get a CCW permit. The first one I got, I walked into the county sheriff's office, filled out an application, got fingerprinted (and those prints were not put into the then-equivalent of AFIS), and walked out with a permit. There are still jurisdictions where all you have to do is ask and not have a criminal record. I know people who do not have records that I would not give a sharpened pencil to, let alone a firearm.