If the router doing the cutting is a normal hand tool (not a spindle controlled by a VFD) then you'll need to run the logic-level signals (usually about 5v) coming out of your parallel port or motion control card that go high or low on a software command (usually M3/M5) through the breakout board and to a relay that can turn your mains power on and off. You probably have a breakout board already in your system with a relay like that, but it's hard to say if it will work with all the new components.
Breakout boards sometimes have a way to control a VFD by generating a range of voltages (usually up to 10v) that correspond to different speeds. But your setup won't respond to that; all you'll get is on and off.