Pets are NOT permitted on any park trails. They are allowed in Juniper campground, the parking lot in front of the visitor center, and in the picnic area across Frijoles Creek between the "No Dogs Permitted Beyond this Point" signs. All pets must be under physical restraint while in the park. Remember, pets may not be left unattended except inside a vehicle. Please be sure animals are provided adequate ventilation. Pets should never be left in a closed vehicle during hot weather. Pets are not permitted on the shuttle bus from White Rock that runs in the summer. Service animals are allowed everywhere their owners go. Identification on service dogs is very much appreciated (so other visitors don't wonder why your dog can go on the trail but theirs can't) but not required. More information on service dogs can be found here.
There are areas on Department of Energy land near the park that do allow dogs. These areas are located between the town of White Rock and the park.
Printable map of DOE lands where dogs can walk. (3.67 mb PDF)