James Tarkowski and Keirnan Dewsbury-Hall were on target as Everton picked apart Burnley on Tuesday night in a 2-0 win.
It was the 18th defeat of the Premier League season for Burnley, leaving them eight points from safety with nine games remaining as fans grew frustrated.
Burnley shared a video of Parker's post-match interview on X once the game had concluded and supporters made their voices heard.
As reported by the Daily Mail, One comment read: “Sack him. Strip the squad. Take time to replace him with someone with bigger b******s. Rebuild. Go again.”
Others read:
"Hide this reply if Parker should be sacked."
"We should sack Parker. Hide this reply if you agree!"
“Get Parker out. That was embarrassing.”
Burnley hid such replies along with dozens of others, prompting one user to write: “Ladies and gentlemen, we got em.”
Burnley's tally for goals conceded stretches to 58, with no other side in the Premier League this season suffering more goals with even Wolves taking in just 52.
Parker knows Burnley must improve
Speaking in the interview, Parker admitted that his side were far from perfect and must start to improve.
"We fell short, we fell well short in terms of everything about our game. There was nothing in our game that was remotely encouraging or pleasing really.
"And at this present moment in time, you know, we're trying to work out what the reasons are.
"I just thought we looked very, very flat. We were poor on the ball and never really posed any sort of problems and at times their quality was obviously far superior really.
"So, a really disappointing night for us. I don't know whether it was the weekend, maybe it took a lot out of us in terms of the general effort and the physicality of what that game was, the emotional state of that game. But for sure we looked way short tonight."