Old fool's party not finishing late (6)
I believe the answer is:
dotard
'old fool's' is the definition.
(an old person whose mental faculties have declined)
'party not finishing late' is the wordplay.
'party' becomes 'do' (do can informally mean a party).
'not finishing' means to remove the last letter.
'late' becomes 'tardy' (synonyms).
'tardy' with its last letter removed is 'tard'.
'do'+'tard'='DOTARD'
(Other definitions for dotard that I've seen before include "Frail old man" , "Oldie suffering" , "Person old and feeble" , "Senior citizen" , "Old fogey" .)