The gauge lights and rear running lights tag light are not working

The first thing you should do is check the fuses. It's possible that a fuse has blown, causing the lack of power to the dash lights, gas gauge, temperature gauge, oil pressure gauge, and instrument panel lights. If the fuses are all good, then you may need to check the wiring harness for any loose connections or damaged wires. If the wiring is all good, then you may need to replace the instrument cluster.

If the front lights work, splice into that wire and and run it back to the rear. They all run off one fuse. No need for a switch.

Check all the related fuses to the rear lights. Also, check the wiring on each bulb, if accessible.

You said same socket, so it couldn't be a bad ground or then neither light would work. And you also said there is no power on the tail light wire at the socket? There should be voltage on the wire with park lights on. Find the rear tail harness. Somewhere there at the rear of the car the park light wire (the tail lights, running lights, all the same) will splice off to both sides of the car. You have passenger side lights, so you know the signal is good that far. Somewhere between the splice and the driver's side, the power is lost, maybe right at the splice. So find the harness, it will be either under the car along the frame, or inside the trunk. That's an old effin car, Howard. Are you sure you want to nurse it back to life? lol. Buy a Haynes manual for it - it will save you money in repairs. Good wiring diagrams, too.
Are those gauges in the instrument panel, and not aftermarket? If they are in the dash, I may know what it might be. I have an '87 Sentra. Fuel gauge, temperature gauge, and warning lights, and turn signals would only work intermittently. Car would start and run, but those things- sometimes not, or they would come on way after down the road. I traced it down to a faulty ignition relay in a relay block above the instrument fuse panel. The run wire out of the ignition switch trips this relay on, and it sends power to a portion of the fuse panel that has the fuses for the gauges and turn signal lights. Replacing the relay was the answer-they always work now. Your gauges may be similar, but almost as likely a loose connection somewhere.

Maybe you didn't reconnect the ground for those lights. Check for tail light fuse or IP lights fuse. You may have to get a test light or voltmeter, and start looking for where power is lost. Start at the rear light sockets, and work back to the front, to the park light switch.

with no dash lights check the bulbs inside the rear of your dash.if bulbs ok . then send it away to a dash repair place found online.

some thing not grounded proplery

10 amp fuse under steering column that says it is for 'Gauge' fixed this problem. That fuse had blown.

Check ground wire connection from rear light harness connected underneath body in the rear.Sometimes the bolt that grounds the wire to the body of the truck corrodes

check your fuses also check the power distribution box under the hood there are fuses there to make sure you don't have any blown bulbs it might possibly be your switch you can check it with a continuity meeter hop this helps