Since school has started, you might try Dockweiler State Beach in Playa Del Ray. Space may be available if you are looking at a weekday. Los Angeles is NOT a very RV friendly town. I can't think of a single place near the center of town where you can camp or dry camp. This topic has been brought up many times be people who wish to camp near one of the major hospitals during a family admission. Not far from the LA central area is Valencia, with Castaic RV park and Travel Village RV park (near Magic Mountain), both are very nice. It would be at least at 45 minute drive to downtown, during rush hour it will take 2 hours (or more). But if you plan your day to start after 10AM the drive is ok.
Good Luck, and Happy Trails.