Well done CLB! (I had a brain fart last night. I've actually been through this system myself. And going to have to again, because mine is missing the plastic wedge.)(Don't think the descriptive word, "pawl" ever actually stuck in my head, though.
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Oldiron... Check first for what CLB said. A reversed pawl. If the pawl is in the right position, then check for that plastic wedge. (I wouldn't imagine it was still there, if the pawl was flipped.)
It's possible that like mine, yours is missing the plastic wedge. Mine engages the brake, then comes right back up, as well. All new cables. All new brakes. And all new hardware in the rear.
MY problem started when I had a shop replace all the brakes. They couldn't get the new parking brake cables on. The front cable wouldn't go back far enough. When I brought it home, I got the front cable to come out further, by working the ratchet several ways, until it did. Took it back. They replaced the cables, now it works like yours. I'd imagine my pawl was just frozen in place. Locked in. Now, of course, it needs that wedge, to be able to hold again.