A quiet member of the family, that's evident (8)
I believe the answer is:
apparent
'evident' is the definition.
(synonyms)
'a quiet member of the family' is the wordplay.
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'member of the family' becomes 'parent'.
'a'+'p'+'parent'='APPARENT'
'that's' is the link.
(Other definitions for apparent that I've seen before include "Obvious; seeming real" , "Evident or seeming to be" , "Visible - plain - presumable" , "Seeming true; obvious" , "Obvious, clearly understood" .)