Might be caused by a badly worn or faulty ignition switch. When you turn the key to start the engine the ignition may be on, but when you release the key after the engine starts the ignition may be turned off. Check to see if you have 12 volts reaching the igniter when the ignition switch is turned on. If you do, the try to start the engine and check for voltage at the ignitor again as soon as the engine shuts off.
I'd also suggest trying to start the car with the transmission in neutral vs. in park. Sometimes the transmission interlock switch designed tol prevent the engine from starting in gear can get sticky or go bad. However, if this was happening I would not expect the engine to fire up at all.