Can I specify template to use with the_permalink?

The the_permalink() function links to the single post view for the given post-type. As per the Template Hierarchy, the page used to render the single post view is single.php.

So, if you want to modify the template used to render the page when clicking the_permalink(), create/modify single.php, or for a specific post-type, create/modify single-{posttype}.php.

If you want to reference the referring page, you may want to look at hooking into template_redirect, and referencing $REFERRER, or something.