68 - Backup light not working?

The two green wires (light and dark) are for the reverse light switch and appears to be connected. The others, would be for the neutral or park safety switch to allow the vehicle to start only in either one of those positions. One terminal is left vacant by design. The switch may be bad. Checked the bulbs at the rear? Fuses? If those check out, you can disconnect the two green wires and jump them together with the ignition on the reverse lights should illuminate. If not, all points lead to a bad switch. Very common. There is a plastic tab that is notorious for breaking especially with age.

Fairly cheap to replace. 1968 CHEVROLET C10 PICKUP 5.0L 307cid V8 Neutral Safety Switch / Range Sensor | RockAuto

Also look into the grounding part of the taillights and backup lights assembly/mounting housings as these 50+ year old trucks suck with there grounding system being all electric switch's and any thing that relies on the bodys metal for ground when the metal has RUST that can be a problem for the ground connection - - - ask me how i know. I have on my 68 C10 a complete separate grounding system from anything that requires a/the item needing a ground.