When the dash is powered off current will be flowing from the input pin of your Cartek device through the 10k pullup of the aux circuitry which ends up trying to power the 12v rail of the dash.
The current would be very low, maybe 1mA but it must be enough to switch the transistor.
Ideally the dash and Cartek device would be powered by the same circuit or at the same time to avoid this situation.