Old- timer's goat destroyed by this medical specialist (13)
I believe the answer is:
dermatologist
'this medical specialist' is the definition.
(dermatologist is a kind of medical specialist)
'old-timer's goat destroyed' is the wordplay.
'destroyed' is an anagram indicator.
'old'+'timers'+'goat'='oldtimersgoat'
'oldtimersgoat' is an anagram of 'DERMATOLOGIST'.
'by' acts as a link.
(Other definitions for dermatologist that I've seen before include "Skin disease expert" , "medical consultant needed" , "Skin consultant" , "Expert in skin disorders" , "Medical expert specialising in skin" .)