Hello and welcome to Just answer.com, Super mechanic here. About your vehicle.
That could very well just be a bad transmission range position sensor, (neutral safety switch) Here is a diagram of the starting circuitry along with a test for it.
Oh..I thought that meant the BTSI, (park interlock) for the trans brake interlock, my bad.
There are not any other components in the starting system circuit. You will want to see where the power is getting lost, (into or out of the trans range sensor) and that the connections are all good.
ASE MASTER
ASE Master tech. 30+ years mechanic, shop forman, service manager