There is a spot on the right about two thirds of the way up to the skylight area that is flat and out of the way. I have even seen a porta john here before. Good luck!
I've been living out of my car in ouray for the past month and have parked at the amphitheatre campground just south of town, the bear creek trailhead just south of town, some backroad just north of town, at the campgrounds and pullouts on camp bird road, and random spots on Red Mtn Pass when it isn't snowing.