My guess would be the Reverse lights and "S" wire on the Starter relay have a short together.
I assume an Automatic trans, because the "S" wire and the Reverse light wire are both together on the trans shift switch.
Automatics have a Neutral Safety Switch(NSS) mounted just above the shift linkage.
The "S" wire is the smaller wire on it's own smaller post on the Starter Relay(solenoid).
It runs from the ignition switch, thru the NSS switch and to the "S" post on the Relay.
When you turn the key to START, 12volts goes out of the ignition switch to NSS switch on trans, if trans is in Park or Neutral, that 12volts goes thru NSS switch and to the "S" post, this closes the relay and Starter Motor activates.
On that same NSS switch is a constant 12volts, when you shift trans into Reverse that 12volts is sent to Reverse Lights and they light up.
I would guess that the NSS switch is where the problem is, when in reverse that 12volts is also being sent to the "S" post.
Could also be shorted wires there, reverse light OUT wire shorted to "S" wire, have some one look at the reverse lights when you crank engine over, if they light up even a little wire is shorted.
If you have a Manual Trans then there is a Clutch pedal IN safety switch, no where near Reverse light switch on trans, so have to be shorted wire, probably at the back of the engine above bell housing
Last edited by RonD; 09-26-2014 at 09:29 PM.