Horse getting old on drink (5)
I believe the answer is:
pinto
'horse' is the definition.
(pinto is a kind of horse)
'old on drink' is the wordplay.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'on' is a charade indicator (letters next to each other).
'drink' becomes 'pint' (I've seen this before**).
'o' after 'pint' is 'PINTO'.
'getting' is the link.
(Other definitions for pinto that I've seen before include "A piebald pony" , "Mount" , "Piebald horse (US)" , "patchy horse" , "H [HORSE]" .)