Old-fashioned parent swallowing tea endlessly (5)
I believe the answer is:
dated
'old-fashioned' is the definition.
(dated can mean old-fashioned)
'parent swallowing tea endlessly' is the wordplay.
'parent' becomes 'dad' (dad can informally mean father).
'swallowing' is an insertion indicator.
'endlessly' means to remove the last letter.
'tea' with its final letter taken off is 'te'.
'dad' placed around 'te' is 'DATED'.
(Other definitions for dated that I've seen before include "Antiquated" , "not without romantic engagements" , "Outmoded" , "square" , "Obsolete" .)