There are some places that even some law enforcement officials are not allowed to be armed. But referring to National Parks, however, each park has its own special regulations, while there are also national regulations as well. Each place has its own situation and its own problems and, in theory, writes its own rules to cope. Some even have no gates and no admission fee (if you drive in).
There are some places that fall under the National Park Service that are neither fish nor fowl, so to speak. There are national scenic rivers, national trails and national monuments. I don't think the rivers have any sorts of rangers and I doubt few trails do either, although the Applachian Trail goes right by Camp David over in Maryland, just over the hill from Greenbrier State Park. But that's a special case.
One thing you might want to remember is that there are all sorts of overlapping laws and regulations and naturally, law enforcement agencies. It is easy to run afoul of something, so you have to be careful. This is true of all laws, of course. Laws can be written so they cannot be obeyed but that usually doesn't happen here anymore, but still! For instance, game laws could be construed to apply to anyone being armed in the backwoods, depending on whose land you were on. National forests have their own rules and as I mentioned in another thread, a totally different outlook on life. Then there are state parks and forests and even county parks, all with their own rules. Finally, there are private places that are probably a law unto themselves.
All the same, in the places in Shenandoah where I usually go (and go less frequently as I get older), I don't see many people at all. Granted, I do see some most, but not every time I go, even on cold and snowy days, but if you run into someone two or three miles from the parking lot, chances are they are dedicated hikers and campers. In 30 years of going there I have never run into anyone the least bit suspicious looking, by which I mean no more suspicious looking than I am. Also, in all that time, I have seen rangers on the trail a grand total of twice, women hiking alone about five times, groups that included children about the same number of times and someone hiking with a dog, about twice. Bears, with cubs, twice. Deer, about a hundred, give or take.