Implicating umbrella_cockatoo → cockatoo
Link to implication

Reason:

A species of cockatoo

Related parrot implications

• major mitchell's cockatoo → cockatoo
• sulphur-crested cockatoo → cockatoo

• blue-and-yellow macaw → macaw
• hyacinth macaw → macaw (remove bird implication; macaw implies parrot)
• spix's macaw → macaw

• african grey → parrot
• conure → parrot/parakeet ³

Related parrot aliases

• budgie → budgerigar ²
• grey parrot → african grey
• major mitchells cockatoo → major mitchell's cockatoo ¹
• pink cockatoo → major mitchell's cockatoo ¹
• sulfur-crested cockatoo → sulphur-crested cockatoo
• white cockatoo → umbrella cockatoo ¹

¹ Aliasing into proper name or aliasing into an apostrophe (see: five nights at freddy's)
² Mentioned here
³ Since parakeet's are classified as "small parrots with long tails", many classify the conure as such. I'm unsure, so I suggested an implication to either.