We employ URL caching by means of a CDN provider. When a registered user is logged-in to the site, he/she is displayed with the pre-cached, non-personalized page.

In order to have the CDN provider distinguish the URLs, we need to add a non-SEF parameter in the SEF URL, eg of the form www.mysite.com/?time=13213548 where ?time=13213548 the current time in UNIX epoch format.

How can we add a similar non-SEF parameter only for logged-in users? We use Joomla 1.5.22 and JoomSEF 3.7.5

Hello,

if this should be added only for logged users, you would have to hack the JoomSEF code and add the needed condition. (if you need this only for specific component, the hack could be done in the extension file, which may be easier, otherwise you would need to hack the core).
If you want assistance with it, you can book a paid support.