IANAL, nor a politician, but IMHO the furloughs are not about saving money.
They are a result of the federal government not having authorization to spend any money.
It's like a company in bankruptcy proceedings, the curator takes over and protects the assets while working to get the best outcome for the creditors.
"these are facilities that don't have any services being discontinued"
If that were true, nobody would be unhappy with their closure, and those places wouldn't be a very safe place to be even before the government shutdown (no maintained roads and trails, no and safety equipment, no animal control/fire/law enforcement/first aid service, etc).
What it's about is both preventing damage to assets and preventing spending of any money not deemed absolutely essential, which they have been instructed to do from the top down.
If a website needs a security update for a zero-day exploit, or gets hacked or vandalized during the furlough, the IT guys are not allowed to do anything about it because they are on furlough. They are not deemed essential employees and therefore they can not do work, any work, including volunteering to support the website (nothing they can do about that, in fact they can get in trouble for breaking those rules). We should be lucky that there is a webpage with a notice: They could have simply powered the machines (cloud, whatnot) off. What if the air conditioning turns off and the server room overheats, or there is some kind of water leak in the room, damaging the running server(s)? If I was responsible for an Internet-exposed website, and I was instructed to protect the assets with only absolutely essential expenditures, and I would be guaranteed not to be able to do anything for it for an indefinite amount of time and there was nobody willing and able to take on the responsibility during my absence, I would shut it down too, to prevent being faulted for anything happening to it in my absence.
If inside a national park an accident or crime happens that needs for example a road closure, a rescue, a fire department t respond, or an arrest (for example, for damaging public property, public intoxication, etc), then the government can't help and can't control the damage because there is no authorization to spend any money to pay for the work and materials of the rescue, fire control, etc. So the best way to prevent damage in the park is to completely close access to it.