That's funny because:

IGN: Is there any chance we could get a Wyvern for the Dragoon Job class? As someone who really loved the Dragoon class from FFXI, I'd really like to have the option of a Wyvern in FFXIV. If not for multiplayer, then for solo play.

Yoshi-P: We receive many similar questions. Probably we could bring you the pet Wyvern anytime in the near future if we wanted to, but at the same time if we want that Wyvern to have the ability to attack the enemies along with the Dragoon then that means we need to reduce the strength of the Dragoon itself because we need to make sure there’s a good balance in the party. So I’m sure that’s not what players want – they wouldn’t want us to reduce the strength that the Dragoon class has already. So as long as the Wyvern doesn’t have that kind of special power then I’m sure we could implement it, so sometime in the future maybe you will see the pet Wyvern. We do have some plans and when you see how the story evolves around Reborn it might make sense to you

http://www.ign.com/articles/2014/05/...tions-answered