Well, I've made legal mistakes before; but, this is the way I understand things; and, if I'm wrong again, I'm certain someone will be along shortly to straighten me right out!
Here's the way I understand it: State roads which traverse national parks may or may not come under federal jurisdiction. If they do, then, all federal rules and regulations apply. Charges may be brought against you before either a: municipal, county, or federal judge for any violation of park regulations which - while not necessarily laws - carry the full force of law.
I live on the border of a large national park. (Delaware Water Gap National Recreation Area) State Route 209 runs, right, through it. Worse, Route 209 is the only major north-south route in the area. If you're traveling from Stroudsburg to Milford, then, you've got to use Route 209. For the length of the highway that traverses the park, Route 209 is NOT a state road; instead, it comes under the auspices of the National Park Service.
Since the last Clinton administration things haven't been very good for people who carry firearms in national parks. Once inside an NP, you're not so much covered by law as you are by fiat. If there's a sitting president in Washington who is strongly anti-Second Amendment - like one of the Clintons - you can expect a whole new, 'crap pile' of fiat legislation to be leveled against your Second Amendment rights whenever you're on NP lands.
So, what's the right way to travel from Stroudsburg to Milford? Remove your sidearm; empty, both, it and any magazines; store them inside a locked box; also keep the ammunition separately locked away, too. Having had to do this a few times, I can tell you that it feels like someone is preying upon legally armed travelers; but, that's the way it is; and, sadly, you may expect things to get worse once Hillary regains control of the White House.
Still, any federal penalties are far less severe than if you make that other great, 'transportation mistake' that dogs travelers in our area. Whatever you do, DO NOT make a wrong turn and inadvertently cross the Delaware into either New York or New Jersey. If you get caught with your, otherwise, legal carry piece in New Jersey, you can expect to spend a minimum of $8,000.00 on court fines and legal fees.
If, however, you get caught in New York state - where it is illegal to import any out-of-state pistol - then they will summarily execute you and your family, right there, on the spot, and hang your lawbreaking carcasses along the side of the road for the crows and buzzards to consume.
(Or, at least, that's how I understand things!)