I spent some time at Voyageurs National Park this fall. In the park, we canoed by 2 other people in 4 days. If you don’t want to camp and canoe you could look into renting a houseboat.
I’d suggest both Big Bend NP and Guadalupe Mountain NP, but the ‘no camping’ is going to make this tough. Not impossible, but the best way to see these are to camp.
I was also going to mention Big Bend. It is out there though.
I’ll second Canyonlands and Capitol Reef. They are the more obscure parks that are close to more spectacular parks. They are kind of a hassle to get to, which makes them even less well known. Hell, I’d say most people don’t even know they exist. When you are in a state with Arches, Bryce Canyon, and Zion, it’s easy enough to fall between the cracks.
Yeah, you have to want to go to Big Bend…it’s not a place you visit casually. There is a place to stay right in the middle of the park, though; The Chisos Mountain Lodge. The Gage Hotel in Marathon is pretty cool, but it’s pretty far away to use as a base camp.
If you have a high-clearance/4WD/AWD vehicle, it’s pretty easy to get away from the crowds in the southwest. There are lots of usually-passable roads in the area around Capitol Reef that have some really amazing scenery.
I’ve been to almost all the National Parks in the US (except Alaska). You can find quiet places with few people in all of them. The majority of National Park Service areas are actually not National Parks, but things like National Monuments, National Recreation Areas, National Historical Parks, etc. They are often just as nice as the National Parks, but on a smaller scale. Out west, they are often huge and little visited. Some of my favorites are Chiricahua NM in Arizona, Natural Bridges NM in Utah, Lake Chelan NRA in Washington and Scottsbluff NM in western Nebraska.
Dry Tortugas NP is considered the most geographically isolated park in the lower 48 states, but not the least visited.
You might like this site for the extensive visitation statistics. Yopu can see a list of visitation for all the NPS areas in 2–8 here. (I can’ link directly, so select 'Ranking Report" and choose “2008”. Aniakchak National Monument in Alaska is the country’s least visited NPS area, with only 10 visits in 2008. I know a few people who have been there and who can attest to the great difficulty of getting there.
Other areas that often are equivalent in scenery and beauty to National Parks are run by the US Forest Service, US Fish and Wildlife Service (which, amongst other things, runs the National Wildlife Refuge System of which there are hundreds of sites nationwide), BLM, and of course state governments.
I could camp but my wife does not want to. I know there are other places besides NPs that are really nice too, that’s why I asked about similar places.
Another thought…
There is a small coastal hamlet by the name of Trinidad that is a short drive from at least 4 State Parks, one National Park, and a National Forest in Northern California, almost to the Oregon state line. It has some very nice bed and breakfasts and other lodging available.
I particularly recommend Prairie Creek Redwoods State Park, and be sure not to miss Fern Canyon.
It’s way away from what everybody else suggests, but Badlands National Park in South Dakota is my absolute favorite. Nobody’s there, you can climb all over all the insane rock features as long as you don’t steal any fossils, and it’s a strange, unearthly landscape.
ETA - I suppose the national park you’d least like to visit is the Washington Mall? 
Aniakchak is one place I never got to up there. It’s very remote and located in a volcanic caldera that is not all that long extinct (about 1931). You land in the crater with a float plane and raft out through the rift in the crater wall. What a rush that must be. The crater was first explored by Bernard Hubbard, S.J. back in the early part of the 20th century. When he first saw it, he described it as an “eden”, with a beautiful lake and lots of green growth and wildlife. A few years later, it erupted and created the devastation you see there now.
Seconding!
I’m biased because I live here, but I have to say this entire area is quite beautiful and there are no real crowds. See this map to get an idea of just how much is around here; the magazine 101 Things to Do (in Humboldt) lists a bunch more non-governmental attractions.
The main draws of this region (for non-campers) are the pristine, mostly empty beaches and the thick redwood forests. The world’s tallest tree, subject of the book Wild Trees and National Geographic’s Oct '09 cover story, is located in this area (though its location is not publicly revealed).
Trinidad and its surrounding beaches/parks are beautiful and definitely worth visiting. The town itself is tiny, but it hosts a few crafts shops and an excellent view of the ocean. One bed and breakfast is located just across the street from a tiny historic lighthouse; there are steps leading down to the beach right next to it.
Just ten or fifteen minutes north of Trinidad is Patrick’s Point State Park, a small but nonetheless interesting site with redwoods, multiple beaches – some sandy, some rocky, some black with agate – a recreated Indian village, a native plants garden, and some rock formations with a commanding view of the bay. I believe gray whiles are migrating in this area right now as well. At night, when there’s no fog, there is so little light pollution in the area the stars are unbelievably clear.
If you’re going to see Fern Canyon and the other northern parks, though, you can also consider staying at the gorgeous and cozy Redwood Hostel located an hour north of Trinidad. It’ll put you closer to the parks than Trinidad.
I could go on and on and on… came here once and fell in love so much I moved here 
I’m biased because that is where I’m from. 
And we’re moving back in a few weeks. w00t! We could do a Humboldt Dopefest, but we’d have to fly in enough people to reach critical mass for a party…
ETA: I’m pretty sure my dad (no longer with us) and my brother were on the last crew to paint and clean the Punta Gorda lighthouse, which is on the cover of the 101things issue.
(Missed edit)
I should add, though, that there aren’t “main” attractions here, no one thing that stands out like Crater Lake or the Grand Canyon; rather, the area is a collection of diverse beauties intermingling in their own little dance.
If you want your trip to be about One Big Thing, you probably won’t find it here, but IMO the hundred little things certainly add up to more than make up for it. This such a far cry from the more popular parks that it results in a completely different state of being; you spend your days here feeling relaxed and rejuvenated, not constantly dodging tourists and worrying about parking and jumping from one overcrowded lookout point to another. Instead you take long walks through the woods, wade through the beaches, and climb a few small rocks to catch the sunset… all in a day’s work. It’s soul-cleansing, really.
A Humboldt Dopefest? Isn’t that… like… every day? 
Seriously though, I never knew there were other Dopers in the area. Shoot me a message or something when you get back!
A Humboldt Dopefest? Isn’t that… like… every day?
Seriously though, I never knew there were other Dopers in the area. Shoot me a message or something when you get back!
Got a call from my parents today.
They’re in Vegas for the National Finals Rodeo and sidetripped out to Zion. Said it was gorgeous, with about 6 inches of snow.
Glacier is awesome. Spectacular scenery, tons of wildlife, but cold by my standards even in July.
I’ll vouch for Big Bend and Guadalupe peak, and second the motion that “no camping” would make them tough. Though Big Bend does have some nice lodges in the middle of the park that would give easy access to most everything you want to see.
Lassen Volcanic National Park has as its unofficial motto The one and lonely. The main road though the park is closed in Winter and Spring until the can get it open. One year we went there on the 4th of July weekend, and the road had just been opened within the week.
Not sure about hotels, though, we went camping.
As long as you seem to be kind of doing this area, might I recommend:
Kasha-Katuwe Tent Rocks National Monument
It is between Santa Fe and Albequerque. It is probably a 3-4 hour visit, but it is truly amazing and we didn’t see another person (other than the entrance guard) while we were there. It is like a mini canyon that you can explore and hike to the top of and then look back down.