Passenger side door & engine power on

I have a consistent problem when I turn off my 2004 Prius. I park, press power the button to power down. Exit from the drivers side, walk around and open the passenger side door to grab a bag or something else, and the entire vehicle including engine turns back on. This also happens if a passenger exits the car after I have fully exited the car. The dash, radio, AC and engine pop back to life!

On accident, I have left my car idling like this for days at a time (until it ran out of gas). Today I only noticed it because my catalytic converter was stolen. I listened to a car with basically no exhaust system starting up every ten minutes or so from my office all day. I even commented to a coworker, "Hear that... that's exactly what my car sounds like"

Has anyone experienced this problem before? Friends and family have told me I am crazy and must be distracted and forgetting to turn it off. However, I have reproduced the issue multiple times and in the presence of others...

I cant imagine this is a safety feature as it engages the engine (still in park) without depressing the brake pedal.

Anyone have any ideas?