National Parks AND National Monuments are off limits to carry and even transporting.
I found out the hard way, several years ago when Halleys Comet was coming close to earth, and I was an astronomy student at the U. of Arizona I took the wife and infant son to the Suguaro National Monument outside of Tucson. I had a telescope set up on the roof of my car, and cameras too. I also had my Mdl 29 S&W on,, as there are often some BGs hanging out there at night. About an hour into observation, the 2 Rangers pulled up for a general checking out, drove away and came back a few minutes later, one came up and asked if I had a gun, I said yes, next thing, the other ranger has a shotgun pointed at me, and tells me to put my hands on my head, they take the revolver and cite me for carrying. They also said they were in within their rights for citing for poaching. I was pretty POed, as it was obvious what I was doing, they could have just told me it was illegal to carry and to lock up the gun in the trunk, paid a $25. fine the next day and got the gun back.
But these rangers were pretty unprofessional, as my wife had a 20 round magazine loaded mini-14 with factory collapsable stock on her lap in the front seat, there was also a loaded Remington 870, with 20 inch barrel by her side, and she knows how to use both of them. They never bothered to check her out or the interior of the car. If we had really been up to no good, we had them outgunned by a long shot.
It is legal to carry on National Forest and Bureau of Land Management land. It's always a good idea to check what the local restrictions may be and to have a valid hunting license with you so that you're not cited for poaching.