I was hoping to climb a few mountains that have no trails in Tucson Mountain Park and Saguaro National Park West District. I called Tucson Mountain Park. The lady at first said hiking off trail is allowed. She then asked someone else and said it is not allowed. She gave me the impression that she did not know and just made up an answer.
The TMP park rules say that you can only ride bicycles on roads, trails, and in washes (MTB'ing is legal on Tucson Mtn Park trails, but not in Saguaro NP except where designated). But there is no similar language for hikers. http://www.pima.gov/nrpr/geninfo/rules.htm
Use sense and minimize impact because the desert surface is fragile.
I used to ride my mountain bike in that area several years back.
Through my experience, it's best to stay on the trail at all costs. There's several types of
spiny desert plants and cacti species in that area, and their spines can really ruin your weekend if you're not extremely careful.