This could something like a Charge pump issue but without knowing more about the controller and software then it's hard to say.

If you don't know what a charge pump is, It's a feature on the controller or breakout board that watches for a signal from the PC software which it monitors 1000's times per second and if it doesn't see this then it will shut down all it's outputs because it thinks the PC as gone AWOL.

However, it usually works in reverse to what you are seeing, it shouldn't allow the steppers to power up without the PC and its control software running. So it could be that the polarity of the new parallel port is different and the output needs inverting in the software.? What software is it. ?

Another feature that could stop the motors from turning is the enable signal on the drives, if this is connected then it would shut down the drives' motor outputs, however, it wouldn't power them down only stop them from spinning.

My first place to look would be going into the control software and look in the outputs for the charge pump signal, then change the active state to the opposite to whatever it's set to know ie: if set to HIGH set it LOW or vise versa.

Other than this can't think of anything else which would cause it to do what you are experiencing without knowing more about the software and controller etc.