When you get a party invite, it should read it in your log and you should see a gold ? at the top of your screen. It should be next to the ! when you are near the Aerth and your status effects.
When it shows up you should be able to enter the main menu and just how you would accept "Check" "Trade" or "Aerth Crys" there should be one that says Party Invite. Select party invite will give you the pop-up window to Accept or Decline.
You could also just type:
to accept or decline party