I live less than a half hour drive from the Cuyahoga Valley National Park. Ok, so it’s not the Grand Canyon or Denali, but it’s still a real National Park.
How close are you to the nearest one?
Less than an hour to Joshua Tree. About 2 hours from Mojave.
45 minutes from Pea Ridge.
About an hour from Olympic National Park.
I’m about 75 miles from Pinnacles National Park. Going hiking there next week. 
Depends how you National Park, part of the NPS or “___ National Park”
10 miles to the World War II Valor in the Pacific National Monument (the official more-than-a-mouthful-name for the USS Arizona Memorial)
135 miles to Haleakalā National Park (but an island away on Maui)
240 miles to the more famous Hawaii Volcanoes National Park
I just walked my dogs in the OP’s national park this week. I am 4 miles away.
I’m about 15 minutes from Valley Forge National Historical Park. I’ve flown kites, hiked, bicycled, picnicked, etc. there many times. Mostly years ago when it was a state park and used mostly as a local recreational area.
I’m only a few miles away from the Pt. Loma National Monument, with the Cabrillo Lighthouse. Very pleasant little place.
I’m also quite close to the San Diego National Wildlife Refuge, and just beyond that is the Cleveland National Forest.
Joshua Tree is the closest park to me that actually named “National Park.” Looks like about eighty miles away.
Yosemite is about 3 1/2 hours away. Yosemite Village is 120 miles as the crow flies, or 167 by car.
It looks like Lassen Volcanic National Park is actually closer by time, being only a 3 hour drive, despite the fact that its a 20 miles or so farther away distance-wise.
About 88 miles from Houston to the Big Thicket National Park. Twice as long to the San Antonio Missions–for those who prefer culture to mosquitoes & alligators. More than Six Hundred miles to Big Bend…
Mount Rainer is 2+ hours away.
It depends on which direction I go. I live near Saguaro National Park, which has two regions, one on either side of Tucson. The eastern section is 20 miles from Tucson, and the western section is 15 miles away.
Hot Springs National Park is 50 miles from me. It’s not a traditional park. It’s the historic bathhouse district of Hot Springs.
the closest park with camping, RV hookups, a lake and cabins is the Ouachita National Forest and Lake Ouachita State Park. 70 miles away.
Less than half an hour to Olympic National Park.
The nearest proper National Park is 483 miles (Big Bend), but only about 125 miles from San Antonio Missions National Historic Site and Padre Island National Seashore.
5 minutes to Francis Marion National Forest.
I live within walking distance of the Erie Canalway National Heritage Corridor. If you want something that says park in the title, the closest one is Women’s Rights National Historical Park in Seneca Falls which is about seventy miles away. If you want an indisputable National Park, the closest one is Cuyahoga Valley National Park in Ohio which is about 260 miles away.
About 50 miles from Channel Islands National Park, though you have to swim to get there.
160 miles to Yosemite.