Three words for you:
NEUTRAL SAFETY SWITCH and it's related wiring!
The switch may be bad or simply disconnected. The wires for it go to an approximately 1/2"-3/4" square connector kinda near (or at the same place as) where the the wires come through the firewall to the coil, gauge senders, etc. The wires from it go down (or are supposed to go) to the transmission. Two of the wires are for the neutral safety switch and the other two are for the reverse/backup lights. With this disconnected (or faulty/misadjusted) your car will run and start, with a push button switch or jumper at the solenoid/relay but NOT from the key switch. And by the way, the plug will be square with 2 male and 2 female pins "IF" your car is late 67. If early 67, it will be split up into 2 separate 2 pin connectors. If there is no switch or wires to the trans, you can simply use/make a jumper from male to female of each pair. If you check your backup lights it will be obvious which pair is which. Remember that a switch in the "on" position is the same thing as a straight piece of wire.
HTH,
Gene