Petrified Forest National Park does not have campground facilities, and overnight parking is not allowed.
Backpack camping is allowed within the Petrified Forest National Wilderness Area, requiring, at minimum, a 1 mile hike away from two designated parking spots. A free backcountry permit is available from one of the visitor centers. Permits may not be attained in advance.
The following are popular camping options outside the park. The list is not all inclusive, and the National Park Service does not endorse the use of any of the following facilities.
Nearby National Park sites:
Canyon de Chelly National Monument: http://www.nps.gov/cach/planyourvisit/outdooractivities.htm
El Morro National Monument: http://www.nps.gov/elmo/planyourvisit/campgrounds.htm
Chaco Culture National Historical Park: http://www.nps.gov/chcu/planyourvisit/campgrounds.htm
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.
There are private campgrounds throughout Navajo and Apache Counties, in and around Holbrook, Sun Valley, St. Johns, Joseph City, and other communities.
The National Forest Service maintains great campgrounds in:
Apache-Sitgreaves National Forest http://www.fs.usda.gov/activity/asnf/recreation/camping-cabins
Coconino National Forest http://www.fs.usda.gov/activity/coconino/recreation/camping-cabins.
Nearby Arizona and New Mexico State Parks include:
Blue Water State Park http://www.emnrd.state.nm.us/PRD/Bluewater.htm
Homolovi State Park http://azstateparks.com/Parks/HORU/index.html
Fool Hollow Lake Recreation Area http://azstateparks.com/Parks/FOHO/index.html
Lyman Lake State Park http://azstateparks.com/Parks/LYLA/index.html
Don't forget to visit http://www.recreation.gov/ for all sorts of good information and reservations for public lands!