I actually solved this myself, but because I couldn't find any Tealium documentation or anything in the Forum about the Parse.ly tag, I thought I would post it here for others.
This is a helpful page:
We did have to map the Site ID because our Parse.ly site_id is just our "base" domain -- so it doesn't work on subdomains. I just set a new data layer UDO variable (parsely_site_id) to the needed value using a Set Data Values extension, and mapped it to Site ID (which appeared in the mapping list).
Great job @RavingLunatech figure it out! And thank you for posting the solution for all of us to use. Much appreciated!