The trouble is not in your brake lamp switch (which has no way of controlling the two lights independently, only together via a single wire). The trouble is almost certainly with your turn signal switch, which will need replacing to fix the problem. The brake lamp feed goes through the turn signal switch. When you signal for, let's say, a left turn, the feed from the brake lamp switch to the left rear lamp is interrupted, and that lamp is fed instead via the turn signal flasher. That is why if you step on the brake with the ignition off, and move the turn signal lever to the left or right turn position, the left or right brake lamp turns off (no feed from the turn signal flasher because the ignition's off). After several decades' service, the switches wear and the shuttles no longer move all the way where they're supposed to go (or all the way back), causing problems like this what you're describing. The '62-'63 switches have flat 1/4" spade terminals, while the otherwise-identical '64-'66 switches have 1/8" square terminals. The '62-'63 switches #2258 700 tend to command a fortune on eBay, because they were used on Max Wedge cars. I have a brand new switch w/'62-'63 spade terminals and if you promise not to use it on a Max Wedge car, I'll promise not to charge you a fortune.
Send me a PM if you want it.The only other possibility is a bad wire or faulty connection in the line from the turn signal switch to the left brake lamp. Does the left rear turn signal work? If so, it's definitely the turn signal switch's fault your brake lamps don't both work. If not, it could either be the turn signal switch or a faulty wire.