Or can it also run on a domain that is parked on that account?

If it were to run on a parked domain, would there be any drawbacks or complications involved (ex. forum would be sending automatic emails from the main domain as opposed to the parked one?)?

It can run on any and every domain; the issue is what domain you set as your primary; that is the domain that all logins will go through and you could run into cross domain cookie errors.

There is a resolve there, and that's to use my SSL mod over at UBBDev, it'll read the host_name variable via PHP and use that as the domain for your login data.