Telegram admitting dad's qualified (7)
I believe the answer is:
capable
'qualified' is the definition.
(qualified can mean capable or proficient)
'telegram admitting dad's' is the wordplay.
'telegram' becomes 'cable' (cable is a kind of telegram).
'admitting' is an insertion indicator.
'dad' becomes 'pa' (synonyms).
'cable' going around 'pa' is 'CAPABLE'.
(Other definitions for capable that I've seen before include "Competent, having a certain ability" , "'Clever, able (7)'" , "Adept" , "Of a person, competent" , "qualified" .)