Acknowledges vanity -- not the thing with some in Paris (8)
I believe the answer is:
concedes
'acknowledges' is the definition.
(conceding is a kind of acknowledging)
'vanity not the thing with some in paris' is the wordplay.
'vanity' becomes 'conceit' ('conceit' can be a synonym of 'vanity').
'not' is a deletion indicator.
'the thing' becomes 'it'.
'with' says to put letters next to each other.
'some in paris' becomes 'des'.
'conceit' with 'it' taken out is 'conce'.
'conce'+'des'='CONCEDES'
(Another definition for concedes that I've seen is " Admits to be true or yields".)