Joomla question

I have a hosting client that had his site designed in Joomla. The problem that has come up, is that the part of the site that needs to be secured, is actually on another domain which is a parked domain inside the primary domain's hosting space, but with links back and forth between the two sites. It is my understanding, that the module for Joomla which produces both the primary site and the secondary site both use the Joomla core files for dynamic production. That being said, a Wildcard SSL was installed to provide security upon access. However, if one puts in the address of the second site via https://, the red warning page appears, and advises that the address to which the browser goes is actually the primary site. If the address to the secondary site is put into the browser without https:// the site shows up normally, as with any other site. Is this because the Joomla production of dynamic web pages actually produces a virtual version of the secondary site? Or, is the secondary site/domain being treated as a "parked" domain with no files of it's own to point to, as opposed to a subdomain/add-on-domain with it's own Joomla setup?