Many people do in Seattle. I have seen full-blown vegetable gardens in addition to flowers.
You are in California, right? Here is an article on it. Apparently, we are wrong about who owns it.
Excerpt:
“Contrary to popular belief, officials say, the city does not own the parking strip (more on this later). Most cities do regulate trees on the strip and maintain them as staffing allows. In many cases, however, any landscaping beyond those trees—be it grass, flowers or shrubs—is left to the homeowner to plant and maintain.
“Some communities have stringent rules, others don’t,” says Ann Meshekoff, owner of Ground Effects Design Group in Van Nuys. “Some cities demand permits for everything.”
Pasadena and Santa Monica promote turf alternatives in the parking strips, though Santa Monica requires a permit. The municipal code of Beverly Hills calls for grass but allows substitutes with approval from the Public Works Department.
Los Angeles’ long-standing guidelines state that between curb and sidewalk, homeowners should plant only turf, and a revocable permit is required for deviations, says Lance Oishi, a senior landscape architect for the city.”