As far as I know the trails in Palo Duro canyon are on private land and are only open for special events like Jeep Jamboree.
Here is some info I found regarding public roads through Palo Duro that may be a nice drive.
"I'll give you directions to start at the south end of the trail. When you reach Clarendon, turn left (SW) on to FM 2162 (AKA Koogle St.,) Which will be the stoplight between the Trading Post and Allsups. Take 2162 all the way down to CO RD X. This is essentially the start of the "trail". Just follow the road. There are many side roads, but explore at your own risk, all of the surrounding land is private ranch land. That being said keep an eye out for cattle. You will cross several cattle guards, but don't worry, as long as you stay on the main road, you'll be okay.
You'll come up to a ranch house, Slow down as you travel through this section as there are children that live there. Drive over their cattle guard, and into their fenced property (remember you're still on the CO RD.) At this point you will make a Right hand turn (N) on to what becomes CO RD 25. Just follow the road back up out of the canyon. It will eventually Lead to CO RD 21, Which will take you out to FM 2272 (AKA Romeo Rd). Follow this road to the west, and it will Lead you out to TX 207. To get back to US 287, simply follow 207 north, and it will meet up with 287 on the west end of Claude, right next to the Cefco. "
I've read about the Jeep Jamboree event at the Figure 3 Ranch in Palo Duro Canyon, but it appears that the ranch is only open for special events like this.
www.find-off-road-trails.com