Old man very strict — bit of a nag (7)
I believe the answer is:
pastern
'bit of a nag' is the definition.
(part of a horse's foot)
'old man very strict' is the wordplay.
'old man' becomes 'pa' (both can mean father).
'very strict' becomes 'stern'.
'pa'+'stern'='PASTERN'
(Other definitions for pastern that I've seen before include "Fetter-bone of horse" , "'The knee of an horse' (Johnson)" , "Word Johnson misdefined" , "Part of horse's foot" , "part of Arab's leg" .)