Parker was not in the dugout at Craven Cottage on Boxing Day after he was forced to isolate after a member of his household tested positive for Covid-19.
Assistant manager Stuart Gray and first-team coach Matt Wells took charge of the team in his absence and Parker, watching from home, would have breathed a sigh of relief in the final five minutes when Walcott thought he had won it.
But the goal was ruled out by VAR when replays showed Che Adams was just offside in the build-up.
The result means Fulham stay in the bottom three.
Parker is set to be back in the Fulham dugout against Tottenham next Wednesday after returning a second negative test for coronavirus.
Andre-Frank Zambo Anguissa fluffed Fulham’s best chance of a quiet first half when he got it all wrong and miss-hit a shot at the end of a slick team move.
Alphonse Areola then produced a wonder save to tip a James Ward-Prowse free-kick onto the bar and Adams could not convert the rebound.
Fulham had a good chance at the start of the second half when Antonee Robinson crossed for a totally free Ivan Cavaleiro inside the area but he headed over.
Parker would have been relieved a penalty was not given after a VAR check when the ball hit Ola Aina on the arm inside the area.
Shane Long then had the ball in the net for Southampton but he was offside and the game ended goalless after Walcott also had a goal ruled out.