If you haven't already, read this NP web page. There is no camping at Petrified Forest, however a couple of places nearby do allow parking lot boondocking.
https://www.nps.gov/pefo/planyourvisit/camping.htmI was planning a similar trip and realized it was impossible to see and do everything in a couple of weeks in one trip.
If you desire to have the comforts and SECURITY of a real campground, Homolovi State Park, at Winslow not far past the Petrified Forest does have a RV campground. NM interstate rest areas have picnic an overnight areas and NM does allow overnight at the rest areas.
There is also a campground at the Sky City Casino at North Acomita Village, exit 102 in New Mexico.
I plan 500 mile legs per day, and prefer campgrounds that have hookups and I feel like will be secure places.
Charles