So far (knock on wood) I've had no problem parking on the side of the road. The location I gave you has a very wide shoulder, so to speak, and you shouldn't have any problems finding a place to pull well off the side of the road.
BTW, I don't remember if you mentioned when you were heading out there, but from mid May on through summer, stating the obvious here of course, it has the potential to be *insanely* hot out on the dunes. Getting to the dunes 20 minutes prior to sunrise helps a lot as it's (relatively) cooler then, but if you choose to do late afternoon, pre-sunset shooting, get ready for some heat. Do not skimp on water, no matter when you go. Those dunes reflect a lot of energy back up at you, so sunscreen of course is a must.
As for sunrise vs set, I've had good results both ways. Far less footprints if you shoot sunrise, and you want to be there before the sun crests the mountain range and hits the dunes to set up your shots. The light for those first 20 minutes (or so) after the sun peeks up and starts hitting the dunes is magical, and it changes quickly.