The Gunners rotated heavily for the tie and welcomed seven new faces to the starting lineup, but the defence of their crown ended at the hands of a full-strength Saints outfit in a 1-0 defeat.
Arteta has since defended out-of-favour players such as Willian and Nicolas Pepe, who failed to grab their opportunity on the South Coast, and Parlour cannot understand why the Gunners boss did not field his best XI.
Speaking to talkSPORT, Parlour said: "You have to put out a strong side. You know Southampton are in good form, they're an excellent side this season and they were always going to put out a strong side, and I expected that from Arsenal.
"I know they're looking at the league, but the FA Cup is very important, it certainly was in my era, when I played under Arsene Wenger he put a strong side out in most cup games.
"But Arteta rested a lot of players today. I would have played my best team, get in front, and then rotate your players in the second half. First and foremost, you have to make sure you win the game.
"It really baffles me. Clubs in mid-table, teams that can't win the title but aren't going to be relegated, why do they rest players in the cup? I just don't get it!
"It's the FA Cup, the best tournament in the world, but now Arsenal are out, so they have to concentrate on the league."
Arsenal must travel to St Mary's again on Tuesday for their Premier League fixture with Southampton, although they have turned their fortunes around in the league to sit 11th in the table. body check tags ::