In proportion to a parrot roughly (3,4)
I believe the answer is:
pro rata
'in proportion' is the definition.
(I've seen this before)
'a parrot roughly' is the wordplay.
'roughly' indicates an anagram.
'a'+'parrot'='aparrot'
'aparrot' anagrammed gives 'PRO RATA'.
'to' acts as a link.
(Other definitions for pro rata that I've seen before include "fairly divided" , "Proportionally" , "In proportion (Latin)" , "Proportionately based" , "maintaining correspondence" .)