Make Permalink go to specific template

Why not make your link include a parameter to change the way it loads the single.php page?

<a href="https://wordpress.stackexchange.com/questions/25008/<?php the_permalink(); ?>?audio-popup=1">Link</a>

Then in the single.php, you can do something like if ($_GET['audio-popup'] == 1) { include popup-player.php; return; } or similar to make it load that popup-player template instead.

There’s other ways, but this is probably the simplest and suitable for a custom site. If you were doing this in a more generic theme way, then I’d say to set up query vars and rewrites and such, but for a one-off, you don’t need that sort of thing.