In the US, it is a safety requirement that the EPB must be a completely independent system from the regular brakes. Since 99% of the braking systems are hydraulic, the EPB has to be either cable driven (most cars prior to 2010) or electronic. Our is electronic.

 

If you insist on just switching off the car using the stop/start button from Drive in a stationary running position (not advised), then you need to make sure that you keep your foot on the brake after stopping the engine until you feel the brake pedal depress slightly, hear the EPB applied, and know that the system has had time to shift into neutral and actuate the parking brake.