They've got a massive budget for signs; one of their favorites is NO CAMPING NO OVERNIGHT PARKING and another favorite is CAMP IN DESIGNATED SITES ONLY. These signs are everywhere there is enough room to park something.
That being said, this past December, I spent a night in the parking lot of Natural Bridges NM. The lot was empty and snow was coming down fast and heavy. I parked right in front of the no overnight signs. The park staff were working outside the next morning clearing snow, none of them said anything to me. The ranger inside the visitor's center said he saw me come in on camera last night and was glad I stopped in for a visit.