The issue has been resolved.

The following directive worked (for PHP 5.2.5 on Apache 2.2.8):
SetHandler application/x-httpd-php5
In my previous post I reported that I had installed the script on a second host and had problems there, too. The first response to my support ticket on the second host was that the script wasn't even installed. I guess they took a quick glance and saw there was no directory called /links/directory and went no further. After I pointed out to them that it was, indeed, installed and that the admin section was working fine, they figured it out. And presto, the addition of the number 5 solved the problem.
My main concern with this is that it might mean my links site will stop functioning if and when there is an upgrade to php6.