I have a 2005 Park Avenue CE. It has an auxilliary plug just below the driver's seat. I have been plugging my radar detector into this plug. Even if I leave it overnight, it will not switch off the radar detector. I know I have the Battery Rundown Protection option because it stated it clearly on the window sticker. I previously owned a 2004 Buick LeSabre Limited CE and with that car the radar detector went off after 20 minutes.

 

I think that the Battery Rundown Protection only turns off vehicle controlled items like the lights and such. a cig lighter should only work when the car is on, but the aux port should work with the car off...i know that my newer dodge, and my bmw both do as i stated.

 

I know on the LeSabre you could have the power outlet have power all the time or only when the key is on. You just change the position of the fuse in the under hood fuse block but I don't think the Park Ave. has this. I just looked at the wire diagram and it looks like the aux power outlets are hot at all times only.