I haven't been to Palo Duro, but I've been to Caprock Canyon State Park. It was cold and windy when I was there in early February of this year, but I practically had the park to myself. I only saw perhaps a half dozen other people.
I enjoyed my time there. I hiked the trails with my 8x10 and made a few good images of the red cliffs and the dry winter vegetation. There are buffalo there that stick to the flat open areas, for the most part, and there are imported African sheep that climb all over the cliffs.
If you are camping out, there are two primitive campsites that are accessible only on foot by hiking in about a mile. There is also an equestrian campsite (bring your own horse), a couple of vehicle-accessible campsites that have water (sometimes), and one small motor campsite with electricity and water (sometimes) hookups.
The park entrance is near the very small town of Quitaque (the locals pronounce it "Kitty Kway"). There's a diner in town, a place to get gas and a few groceries, but don't count on stocking up on supplies there. Do that in Amarillo, Lubbock, or Childress, depending on where you are coming from. You should plan on bringing your own drinking water in. Water is often in limited supply, and when I was there, there were posted warnings about high levels of some contaminant, but I don't recall what it was. Check with the park rangers about the water.