Bad ignition switch. There is a separate feed from the switch that bypasses the ballast resistor during start. Power flows from the "start" terminal through the coil to the distributor.
Then when you let the key back up to run, the power flows from the "run" terminal through the ballast resistor. So the "run" circuit only is hot during running, and the "start" circuit is only hot during starting.

Replace the ignition switch. This is a very common problem.

R.