Starting the day on 126-2, Pakistan reached 152 before Azhar Ali was run-out by Mitchell Starc for 71.
Nathan Lyon then took three wickets while 39-year-old Younus (136 not out) scored his 34th Test hundred to take his side to 271-8 at the close.
Pakistan still trail by 267, needing 68 more runs to avoid the follow-on.
Rain delayed the start of play by more than three hours at the Sydney Cricket Ground, with only 54 overs bowled as a result.
Younus' maiden century in Australia means he becomes the first player to score a hundred in all 11 nations that have hosted Test matches.
Australia batsman Matt Renshaw - who scored his first Test century on day one - left the field after complaining of a headache following a blow to the helmet when fielding at short leg. Wicketkeeper Matthew Wade also left the field because of illness with Peter Handscomb taking the gloves.
Pakistan's Younus Khan hit an unbeaten century but Australia continued to dominate the third Test on a rain-affected day three in Sydney.