I was doing some county road relation tagging on OpenStreetMap, and apparently Delaware Water Gap NRA isn't the only NPS institution to have numbered roads, as Wind Cave National Park has NPS 5 (
sign)and NPS 6. Like Delaware Water Gap, these are extensions of county routes, but this is western South Dakota and county roads are 1-lane and gravel. They likely don't meet inclusion criteria, but I'm bringing them up solely because they're 2 out of the 4 numbered NPS roads in the country.
Interestingly, Custer State Park has a numbered route network (at the end of
https://www.custercountysd.com/wp-content/uploads/2022/12/CusterCountyAtlas2022.pdf, Custer County seems to be a bit pedantic on route numbering) but that doesn't necessarily qualify as its own system, just thought it was unique.