Parrot with head of myna and a cry of a rook (5)
I believe the answer is:
macaw
'parrot' is the definition.
(macaw is a kind of parrot)
'head of myna and a cry of a rook' is the wordplay.
'head of' suggests taking the first letters.
'a cry of a rook' becomes 'caw' (I've seen this before).
The initial letters of 'myna and' is 'ma'.
'ma'+'caw'='MACAW'
'with' acts as a link.
(Other definitions for macaw that I've seen before include "Chattering cousin of a cockatoo" , "brilliant winger" , "Colourful creature" , "New World parrot" , "Long-tailed American parrot" .)