Father's bird is over the hill (4,2)
I believe the answer is:
past it
'over the hill' is the definition.
(both can mean getting old)
'father's bird' is the wordplay.
'father's' becomes 'pa's' (pa can mean dad or father).
'bird' becomes 'tit' (tit is a kind of bird).
'pas'+'tit'='PAST IT'
'is' is the link.
(Other definitions for past it that I've seen before include "Unable to function any more, perhaps due to age" , "undrinkable?" , "Far from fresh" , "Spit at, being too old or decrepit" , "Informally too old to function" .)