National parks make things sound worse than they actually are to keep the idiots out.
Stock tremor is 100% high clearance - the only caveat is that it is still a full size truck so switch backs and narrow sections can pose a challenge.
This the the crux of the matter.
There is absolutey zero room for debate that a stock Tremor (which is on 35" tired, for crying out loud) is "high clearance". I promise there are no officially-sanctioned roads in the national parks wherein true clearance will be a limiting factor for these trucks.
"High clearance" doesn't mean "small enough to fit anywhere without bumps and bruises". That's an entirely different analysis. But, again, these aren't trails for the sake of being trails... they're access roads for FS and NP staff and for visitors to get to hiking trailheads and backcountry areas. The staff and rangers are almost always traveling them in unmodified vehicles... and rarely with the benefit of 35" tires.
Ground clearance = 100% good to go.
Trail width clearance, especially considering erosion and overgrowth = 90% good to go, but your call. You might need some muti-point turns and take some rocky mountain pinstripes, you'll physically 'fit' otherwise.
I will say that the Maze section and the Squeeze on Elephant hill in Needles are the only places in the UT NPs where you'd really encounter a question of truly being able to fit. The Squeeze absolutely can be done in an SD (mirrors in, mind you) but admittedly does require the right line or you'll scrape. The Maze area has a few snug spots, but a SD will fit, just a little bit of a pucker factor for folks who aren't use to that sort of thing.
I don't think any other NP in the lower 48 will have anything that tight, other than tree overgrowth with is unpredictable based on time of year and road maintenance.