You can't always trust your GPS!
Every year there are several cases of visitors being led astray by their GPS, sometimes with disasterous consequences. And not just GPS but also google maps, mapquest and other directional sites.
There are dirt roads in the area that appear to be the shortest route to your next destination. Yes, it might be the shortest route and will certainly be scenic BUT these roads are not always maintained and can be impassable after a summer rainstorm. Also, you need a high-clearance vehicle for most of these roads. There are NO services on the dirt roads and if you breakdown you can be stranded until someone comes along. Cell service can be spotty. Temps in the summer can be over 100 degrees or in the shoulder seasons a sudden snowstorm can strand you.
Some of the roads most commonly recommended by GPS are:
Cottonwood Wash Road
The Burr Trail
My best recommendation is to stay on paved roads unless you have gotten some local information about the condition and the advisability of the particular dirt road you are considering. And NEVER take a dirt road at night.
All that being said....if you are prepared (food, water, first aid supplies) and a suitable vehicle, these roads provide some of the best scenery in the area and a chance to get away from the crowds at the parks.