From a Patch reader:

When on private property (like BART's parking lot), could you get a ticket for not stopping at a painted word "Stop" on the ground with no stop sign? First Part Answer, BART Police Sgt. Carolyn Perea: BART parking structures are subject to all laws of the road. We do cite for going through a stop sign.

Second Answer from Bradley Sadek, CHP: Publicly owned (state or local) parking structures (lots or garages) are subject to the "Rules of the road." Privately owned lots (most commercial shopping centers) are not subject to the rules of the road.

If you run a stop sign in a private parking facility, you would not be issued a ticket for doing so. However, it does not relieve the driver of their responsibility to safely operate a vehicle. If you cause a collision by running a stop sign, for example, you would still be both civilly liable, and if you caused injuries, criminally liable. Also, there are regulations that prohibit "Reckless driving" in off-street parking facilities, whether public or private, and all DUI related laws still apply.