Type of large parrot commonly kept as a pet (8)

I believe the answer is:

cockatoo

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'type of large parrot' is the definition.
'cockatoo' can be an answer for 'parrot' (cockatoo parrot is a kind of parrot). I am not sure about the 'type of large' bit.

I can't explain the remainder of the clue.

Can you help me to learn more?

(Other definitions for cockatoo that I've seen before include "Crested bird" , "Australian parrot" , "Illegal Aussie look-out man" , "Lightly coloured crested parrot" , "Parrot native to Australia and Indonesia" .)

I've seen this clue in The Times.

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