Parrot many a bird cry (5)

I believe the answer is:
macaw
'parrot' is the definition.
(type of parrot)
'many a bird cry' is the wordplay.
'many' becomes 'M' (Roman numeral for a thousand).
'bird cry' becomes 'caw' (I've seen this in another clue).
'm'+'a'+'caw'='MACAW'
(Other definitions for macaw that I've seen before include "Large bright parrot" , "New World parrot" , "Tropical bird" , "Colourful creature" , "Brightly-coloured parrot" .)