Look up, those are canales
Canales used to be an important part of adobe buildings. They carried water off the roof and through the parapet. You can see them on the exterior of one of Tucson's oldest buildings, La Casa Cordova, 173 N. Meyer, in what is now part of the Tucson Museum of Art complex. The first rooms of that house were built before the 1850s, according to "The Restoration of La Casa Cordova," published by The Junior League of Tucson.
Most canales are plain and simply serve to drain water far away from the foundation of mud adobe walls. But a few have whimsical detail. Most in Tucson are metal, a few are wood or stone. Sadly, gutters have replaced canales on many buildings. The former are no doubt more effective but without charm.