Shut up about father being such a stickler (6)
I believe the answer is:
pedant
'such a stickler' is the definition.
(I've seen this in another clue)
'shut up about father' is the wordplay.
'shut up' becomes 'pent' (synonyms).
'about' indicates putting letters inside.
'father' becomes 'da' (da can mean dad).
'pent' placed around 'da' is 'PEDANT'.
'being' is the link.
(Other definitions for pedant that I've seen before include "Stickler for trivial detail" , "One excessively concerned with minor detail" , "Pentad (anag.)" , "Stickler for minor details" , "quibbler" .)