Tolerate insect, starting as a pet hate (7)
I believe the answer is:
bugbear
'a pet hate' is the definition.
(I've seen this before)
'tolerate insect starting' is the wordplay.
'tolerate' becomes 'bear' (synonyms).
'insect' becomes 'bug' (bug is a kind of insect).
'starting' says to put letters next to each other.
'bear' put after 'bug' is 'BUGBEAR'.
'as' acts as a link.
(Other definitions for bugbear that I've seen before include "Cause of obsessive fear" , "Cause of excessive anxiety" , "Cause of great distress" , "[BOGYMAN]" , "Object of obsessive, foolish fear" .)