1986 Camry parking light wire

I can't think of any wire running to the back of the car that would be 1 volt........I'm not saying this to insult you, but could you be using your meter wrong?  Leads reversed?  Poor ground on the meter's black wire?  Bad meter?  Go underhood and measure the battery voltage, just to make sure you're using it properly and it's still in working order.  Or re-test the brake wire to verify the meter.

I haven't worked specifically on an '86-style Camry, but I've done plenty of Toyotas.  In my experience, you can nearly always find the green parking light wire at the fuse box, and it's nearly always in the same plug as the GREEN / WHITE brake wire.  Or you should be able to find both in the driver's door sill, in the harness that leads to the rear of the car.

Also, another dumb question, but do the car's parking lights actually work?  It's always possible that you blew a fuse during your install, or the parking light circuit has some other problem that you never knew about-----switch them on and verify that they're actually working.