Phillip,
It is very unusual for the clock to run only while engine on. The original clock draws a tiny amount of current from the battery, and that energy is stored in a spring that drives the clock mechanism. The spring is loaded by an electro magnetic servo every few minutes or so. The draw is so small that it would not, imo, be possible to start your car with a battery that was low enough to be unable to drive the clock. So...one of two things have happened. Either someone replaced the movement with a modern movement that requires constant current and/or someone wired the original clock improperly so that it is only hot when ignition is on.
g