For what it's worth, you are not going to come out farther ahead by parting it. You'll never sell every last piece, and probably still fall short of what you consider to be "fair value" for this rig.
If it was me, I would re-evaluate the "fair value" and sell it as a whole to the first person who gives a fair offer on it. But, that's just me, you need to decide what you gotta do
