It's not the fusible link; that would kill the whole car, electrically.
Take a look at your fusebox and see if you have some blown fuses. Let there be a short to ground (for example, one or more of the tail/stop or park/turn light feed wires chafes on the metal edge where it enters the socket) and you'll blow a fuse and lose the applicable lights on both sides of the car. Tail + Park are on the same circuit. Instrument panel lights were on that same fuse in my '65; that was common practice for a lotf years to give you a visual indication at night that you're dark from the rear if something went wrong with that circuit, but it was never a law or regulation or anything, and I think they moved away from that a few years later.
Do you have the factory service manual for your car?