Sky Bet currently have the 40-year-old at 7/4, followed by Slaven Bilic and Ole Gunnar Solskjaer at 4/1 and 5/1 respectively.
Fulham, who won promotion via the Sky Bet Championship play-offs in August, have taken just a point from their opening six games and sit rock bottom of the table.
The Cottagers were last in the top flight in 2018/19, a season in which they sacked both Slavisa Jokanovic and Claudio Ranieri before finishing the campaign cut adrift in 20th with Parker as caretaker boss.
Fulham's latest defeat came courtesy of a ruthless Palace side, who took full advantage of their opponents' weaknesses.
They led in the eighth minute when Jairo Riedewald fired in after being played in by Wilfried Zaha, and the Eagles' talisman doubled the advantage by getting on scoresheet himself midway through the second half.
His fifth league goal of the season came following some excellent play between Andros Townsend and Michy Batshuayi down the right, with the latter's devilish low cross slid home at the far post by Zaha.
Fulham’s afternoon got worse in the final 10 minutes when substitute Aboubakar Kamara was sent off for a foul on Eberechi Eze after referee Graham Scott upgraded his initial yellow-card decision following use of the pitchside monitor
Tom Cairney then scored from distance deep in injury time, but it was nothing more than a consolation.
Palace are now up to fifth in the table with 10 points from their opening six games - a tally like that looks a long way off for Fulham, as even at this early stage they face a real challenge to avoid a relegation fight.
Parker will be frustrated as his side controlled much of the game, but struggled to carve out clear chances.
He will hope to at least be given the chance to lead the Cottagers in their crucial clash with fellow strugglers West Brom on Monday, and their trip to West Ham before the next international break arrives.
Parker 'determined to turn things around'
The Fulham boss has vowed to give it his all to fix his side’s dismal start, having guided the club to promotion last season.
“I can’t control that,” said Parker when asked about whether he feared for his job.
“What I can do is what I have normally done throughout my career and life, during these hard times, it is the time to front up and keep working as hard as I can to get the rewards that I deserve.
“It is out of my hands but I will keep working hard.
“We are a young team, still trying to gel but there were some big positives today and this is when the chips are down you have to face up and keep working hard and I am sure we will come out of the other side,” he said.
“I felt we need to learn from exactly how this game panned out. We had a dominance but at times in the final third lacked a real intensity about us.”
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