My theory is that you get it after your chocobo mastered a role. WHM barding for healer chocobo, BLM barding for DD Chocobo and full plate armor for tank chocobo.

But what I want to know, in the one trailer we saw a chocobo in a Sleipnir barding (Sleipnir = Odin's horse). Do we need to defeat Odin to get it?