It's allowed if you don't ask?
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.