The West London side's winless run stretched to seven matches with defeat at home to Burnley and they can no longer climb out of the bottom three.
Fulham have three games remaining this season and Parker, whose future is uncertain amid links to Tottenham Hotspur, may already be preparing for the 2021-22 campaign.
Mario Lemina will definitely play no part at St Mary's as he is ineligible to face his parent club, and some other loan players may also be overlooked for starting spots.
Bobby Decordova-Reid has been given a regular run in the side and should start again here, even though he has now gone 12 matches without finding the net.
Fulham possible starting lineup: Areola; Aina, Andersen, Adarabioyo, Robinson; Loftus-Cheek, Anguissa; Cavaleiro, Onomah, Lookman; Maja
Watch the Sports Mole Football Shorts prediction for this game below: