Are you doing it in a day or camping? It can be driven in one day, but that is going to be a very long day and I think is a disservice to yourself.
Anyway, the trail is really a double track (mostly) unimproved road. It has been graded at some point in time, but I think the NPS policy is now to only stabilize as necessary. It is a road through a desert environment with a lot of cryptobiotic on the sides, very sensitive to damage. So you are right to ask about pull-off areas. It is frowned upon (and rightly so IMO) for you to just pull off the road anywhere. There are a few established parking areas at very high traffic sites, like Musselman Arch, Airport Tower, White Crack Overlook, Monument Basin, Mineral Bottom Swtichbacks. Any of the campgrounds (you'll recognize those mainly because they have the outhouses) are also good places to park your truck and hike. A few other places will have large outcroppings of sandstone that are fine to pull off the trail onto. There are also a handful of existing wide spots in the trail that are sort of grandfathered in as OK. It's also generally OK to park in a wash, since these are periodically cleaned naturally. Please do be mindful that just driving off the trail may leave tire marks and boot prints in the soil that could be there for decades.
Best advise I can pass along is to ask the Ranger at the Visitor's Center what's up.