Parks in the Southwest...that's a bit like saying, "I'd like a piece of pie...what would you recommend?"
Well, do you want a meat pie? Fruit Pie? Spinach Pie? So it is with parks, both National Parks, National Monuments, and State Parks.
What is it that you're looking for? Do you want to drive to the photo or would you hike for a while? White Sands National Monument is great if you go all the way to the end of the road, get out & hike for a while. (Hint - take a GPS receiver with you - it get's confusing back there.) If you go during a full moon, they'll let you stay out on the dunes until about 10:00 pm. The Monument used to be open 24 hours a day and you could get really nice pre-dawn and sunrise photos - not any more. Now they have "hours."
Bryce Canyon - get out & walk to get the best photos, ditto Arches. Then you get to the Utah parks where you need a good 4-wheel drive vehicle if you really want to get to the exotic stuff + hiking (Capital Reef, Canyon Lands).
Grand Canyon - hard to take a BAD photo if you get up early, stay late and / or weather is moving across. Lake Powell, Lake Mead - rent a houseboat & stay for cruise around.
Black Canyon of the Gunnison....hmmmmm...that's a tough one. Narrow, deep and hard to get a good photo.
Organ Pipe Cactus National Monument, very interesting. Accessible at all hours of the night and day. If you want to go up the Bates Well Road - better have 4-wd.
Carlsbad Caverns - take your time. Take a tripod - take a trip to Guadalupe Mountains National Park.
Most of the fun is the drive. Stay off interstates unless absolutely necessary. Learn to enjoy the process of getting there - not just the end destination.