This is CPanel, right? When you park a domain in CPanel you're doing two things:
Setting up a dns entry - either a new zone or a new A record
Setting up the virtualhost by adding a ServerAlias in the existing vhost container
To get the same effect without manually adding the parked domain you need:
Wildcard dns - for subdomains of the domain you're using (or even all domains if necessary). This is a non-standard setup, but some hosts are willing to do it for you if you only need wildcard subdomains.
Virtualhost settings that will accept any domain name and send it to the same document root. I think CPanel will do this by default if your site is on a dedicated IP. If not, again you could ask your host for a custom setup.
So, ask your host. If they won't do it for you there are others who will, or you could get a VPS or dedicated server where you can set it up for yourself.