I've recently had to investigate and fix mine on my '69, and it was a fairly straight-forward fix. I would imagine that the '65 wouldn't be too much different in the parking brake department.
If you have the shop manuals, they did a great job of identifying everything in the parking brake system.
First off, there's the pedal and release control assembly, inside your driver's compartment. If the pedal's going all the way to the floor, you're not connected somewhere. Follow that cable assembly down underneath your car, and you should see it connected to a cable that stretches in a trapezoidal pattern from the front of your car back to both sides of the rear frame rail in the front of your rear wheelwells.
If that connection is good, look at the ends of the cable in the rear. Ensure both sides should be entering a hole in the front of the brake drum.
If that's the case, you'll need to pop the wheels off and open up your drums. The parking brake cable has a parking brake cable housing retainer that opens up and keeps the end of the parking brake cable inside the drum. The very end of the cable should fit around a notch on the parking brake lever (which is mounted to the inside (toward center of the car) of your secondary (rear) brake shoe. That lever is just a crescent-shaped piece of flat metal that attaches at the top to the secondary shoe, and at the bottom to the parking brake cable. At the top of the lever there is a notch that accepts the parking brake link, which is what pushes against the primary (front) shoe.
There are literally only 4 pieces to a rear-drum parking brake:
Parking brake lever
Parking brake lever retaining clip
Parking brake link
Link spring
Of these, only the parking brake lever are specific to Left and Right.
Again, if you have the shop manuals, opening up and working on drum brakes is very simple and diagrammed very well, even if you've never worked on them before.
Just as an aside, fixing only one side is not like to do much.

I tried this. Initially my cables were not even inserted into my drums, and when I broke them down, I was completely missing my passenger side brake lever, and tried just fixing the driver's side, hehe.
Hope that helps, and hope it was clear enough.
