Is there enough room to park and turn around a pick-up with a 23' camp trailer at (or near) the trail head for White Butte (ND high point)?

Yes. You should be able to park a vehicle that size on the section line. Just don't park so you are blocking the road or driveway to the abandoned yard, as the rancher may need access while you are there. Turning around will likely be accomplished by just backing out into the road. But watch the weather and road conditions. You won't want to be pulling a trailer on Slope County roads after much rain.