Just go to your pet's page and scroll down under the main box under their picture where you will lots lots of buttons. One of those buttons will say "Add Pet to Party", click that and the pet will be added to a party, unless there are no other slots in you party available. If that is the case then you would need to buy a new party slot. To do this, find Esior's Haven from the Map and go into Sylesti Plaza, where a button under William the Hunter will allow you to buy new party slots.
Hope this was helpful, good luck!

At what point can you expand your party? I currently have 2 (lvl 20 & 16) and he's yet to let me add another. Also, what are the rules for pet levels? I picked up two lvl 30's from another who was clearing out their lair (leaving the site I think) and it says I cannot control them yet.

@Calliope,

If I can remember correctly, you can buy a third slot for your party from the Sanctuary of Saerielle Plaza for around 50,000g (I think).

As for the two lvl 30s, you will not be able to control them until your first pet is at that level. (well that is what I have heard)

I hope this helps.