Just replace it.  To remove it from the backing plate, put a worm type hose clamp around all the spring fingers so you can get it back through the hole.  You will have to remove the brake shoes in order to release the cable from the actuation lever.

Assumes you have stock drum brakes.  If otherwise, you are going to have to state what you have.

https://www.npdlink.com/product/cable-parking-brake-front-replacement/148388?backurl=search%2Fproducts%3Fsearch_terms%3DParking%252BBrake%252BCable%252BFront%26top_parent%3D200001%26year%3D1965&year=1965
If it's frayed around the pulley, it may have split.
https://www.npdlink.com/product/pulley-parking-brake-cable/148382?backurl=search%2Fproducts%3Fsearch_terms%3DParking%252BBrake%252BCable%252BFront%26top_parent%3D200001%26year%3D1965&year=1965

At the bottom of the T handle where the cable attaches, is a pulley that the cable rolls over.
When that becomes damaged, it causes the cable to fray.
First link is to the cable you need, second link is to the pulley you probably need.
Edit: assuming you have a 65-66.

Last edited by 50vert (9/07/2020 4:28 PM)

I remember my first car, 1950 Plymouth Cranbrook. Always had to park on a curb since the battery never kept a charge. But that was also the park "brake" since I would always turn front wheels such that if it did roll away it would back into the curb. I could get in the car after school and with left foot out the door roll the thing back and quickly get it in gear and dump the clutch to get it started. Worked great and never rolled away. You gotta remember that was in West Virginia, so nothing was flat. I bought that car for $20.00 ! I finally put an Olds 324 in it with the hydramatic. Then it was a hoot!

Where did you find my picture?