Fulham were 3-1 up just ahead of the hour mark, but Burnley continued to fight, getting a goal in the 86th minute through Oliver Sonne.
Unfortunately for the home side, it was too little too late, and the Burnley fans made their feelings known after the game with boos.
Speaking to reporters after the game, Parker, who got the side promoted back to the Premier League last season, admitted he was heartbroken by the reaction.
“That makes me sad, obviously,” he said. “It was only four months ago that I was standing on the balcony in the town centre and all of us were celebrating and the fans were right with us, and within four months that quickly changes.
“I understand their frustration. This is the world we’re in currently. I get it. How the world of football is, the instant reaction is to boo.
“I don’t think you can question the commitment of this team. This team are foot to the floor and hugely committed and nothing’s changed. The level of hard work, the commitment we put in to be successful last season is exactly the same now.
“It breaks my heart at the end of the game because we came here today wanting to please our fans and wanting them to support us.”