Try it. It varies depending on the locale and the area's supervisor, and sundry other agencies who also get into the mix if they have some agreement with them. Whole process is surprising.
By the time you start getting all the paperwork, liability insurance for about $4-$5 million with the ACORD insurance forms signed over to them, sundry permits, and road encroachment release from whomever built the road in the park (They weren't built or maintained by the Park itself, but likely the local or county Roads or State Roads Dept.) and maintains it for you to park on, the local or county business license for the day, a monitor or ranger that will be assigned to you for ~$85-$110/hr. and weekdays only, possibility of needing a porta-potty hauled in and pumped out too depending on how many involved, and you'll learn something about dealing with the feds. I did it once with the U.S. Forest Service and it ran up to $2,300 for three hours, and the biggest rip off was the county roads department "Roads Encroachment Fee" for parking in a paved parking pullout for three hours which had to be signed off by the Forest Service before they'd sign off on their $230 non-refundable permit which was the cheapest part, next to a one-day business license for that county. Anyone else could park there for a $5 Adventure Pass, but if you want a permit, it's $1,600 for three hours to park in the same $5 spot (I kept their Roads Dept. encroachment parking approval paperwork as it is sort of humorous to look at it all!). It's a total wallet-buster and you'll get the impression they don't want you there flying your drone too, unless you pay through the nose for the privilege.
There is no set fee once the process starts, and it climbs fast from their published rates which you might find on some of their websites if others are part of the permit. It may be non-refundable too once it starts so watch out. As with FAA waivers, they are not fast either and it can take a few months too so plan way ahead. The advantage is with the ranger present, usually a Law Enforcement arm of whatever agency, no one will bother you and you are good to go.
Also, no one at the park itself can grant permission. This can only come from the person in charge of all National Parks.Try it. It varies depending on the locale and the area's supervisor, and sundry other agencies who also get into the mix if they have some agreement with them. Whole process is surprising.