Neutral safety switch could be bad or out of adjustment. Try this:
Put emergency brake on max. Put both feet on regular brake pedal and push HARD. Now, turn key to start with one hand and move shifter around. Try starting in neutral and move stick slightly forward and back. See if engine trys to crank. If it does, turn key to OFF right away. Anything more than this will probably require a wiring diagram. Oh, and make sure no one is standing anywhere near the car when you do this test as the car may move.