past times vs old

past times and old both are nouns.

Word	Noun	Adjective	Verb	Adverb
past times	Yes	No	No	No
old	Yes	Yes	No	No

As nouns, old is a hyponym of past times; that is, old is a word with a more specific, narrower meaning than past times:

past times (noun)	old (noun)
the time that has elapsed	past times

past times (adjective)	old (adjective)
	of long duration; not new
	(used especially of persons) having lived for a relatively long time or attained a specific age
	skilled through long experience
	(used for emphasis) very familiar
	excellent
	just preceding something else in time or order
	belonging to some prior time