Emotes are said to work - try waving at it to attract it to your shoulder. Then defeat an NPC base, find treasure, or defeat an Alpha creature. Apparently you need to be a good pirate. Unsure on the amount and finer points still.

Emote near a parrot it latches onto your shoulder, after this kill Alphas, open loot drops or use a message in a bottle to get treasure chests to gain taming effectiveness