FFXIV 5.5 How to Unlock the Dolphin Calf Minion
To get your hands on this dolphin, you will need to complete an achievement linked to the Ocean Fishing feature.
The achievement you'll need to complete is called "Double-backwards Somersault". This achievement is not something you'll be able to rush as you will be asked to encounter three dolphin pods during your fishing sessions.
The spawn of the dolphin pods is random and you're guaranteed to see one every run so only patience will lead you to the dolphin minion.
If you never participated in Ocean fishing do no worry, you can get on the boat as soon as you finish the intro quest for the fisher job and no score is required to claim your minion.