Parties are temporary groups to run a particular piece of content, and that's what Party Finder is geared for. It's neither designed for nor suited for finding Free Companies.

Some ways to find FCs:
1) Check the forum for the world you're on. For instance the Moogle forum is here.
2) Ask players you've grouped with for an FC invite. (This works if you've grouped locally, like through the Party Finder or with FATE groups. Since the Duty Finder can pair you with players from other worlds, however, it doesn't usually work to try to find FCs from people you meet through DF.)
3) Ask in the /shout channel whether there are FCs recruiting for your particular area of interest. (Casual/Social FCs are different enough from End-game Raiding FCs that it helps to describe a little of what you want from one.)
4) Right click on a player and choose the Free Company Info option. This will bring up a window showing you which FC they're in, what it's main focus is, and whether they're recruiting or not. If they are recruiting for new members, there's also a button that lets you apply to that FC. (Applications submitted this way have to wait until an officer of the FC gets around to checking for applications. It's not instant.)