97 Taurus Intermittent Starting

Car will not start at times. When this happens you can start the car manually by shorting terminals on the solenoid. At least I think that is what it is. Mounts on the right side of the engine compartment. AAA showed me how. Safety switch that prevents starting unless car is in park has been replaced. Used test lamp to determine that when the car does not start there is no voltage coming in to the solenoid. Looking for ideas on what to try next.