We've camped on a spur along that road, but only once. I believe that 3pin knows this area rather well.
.............................................maybe
Traveling north from Bodie on the Bodie Masonic Road there are a few wide spots such as 38° 13' 56.2" N 119° 02' 36.51 W where you can pull well off the road and spend a quick night. Most of these spots are associated with cattle grazing, and I expect that cattle have been put into the Bodie Hills for the summer.
I will PM Lighthawk, Bill, and Bob the coordinates for one of our favorite campsites that we have used just a little further north.