Wayne, Have not camped in the area, but found these on the nps.gov for the Petrified Forest web site
Outside the south entrance of the park, at the junction of the park road and Highway 180, two privately owned gift shops allow overnight parking in their parking lots.
Off Interstate 40, about 18 miles west of the north entrance to the park, Sun Valley has the “Root 66 RV Park” with hook-ups.
In Holbrook, 26 miles west of the park, there is both a KOA and a Good Sam’s RV park (the OK RV Park). For more information about places to stay, contact the Holbrook Chamber of Commerce at (800) 524-2459 or write them at 100 E. Arizona, Holbrook, AZ 86025.
Near Winslow, 1 hour west of the park, is Homolovi Ruins State Park with a campground and facilities.
Near St. Johns, 1 hour southeast of the park, is Lyman Lake State Park with a campground, cabins, and yurts.
Other state parks in the region can be explored at Arizona State Parks.
Near Gallup, New Mexico, 1 hour east of the park, is Red Rock State Park, with a campground and facilities.
http://www.nps.gov/pefo/planyourvisit/camping.htm
Bob