Relegation-threatened Burnley have held Manchester United, Liverpool, Tottenham Hotspur and Chelsea since January in a prolonged battle to avoid an immediate drop after earning promotion to the Premier League last season.
But Saturday's loss, their third in the last five league games, left Burnley eight points below the safety zone, as seventh-placed Brentford moved within five points of the top four.
"The fight and the spirit was incredible... I’m proud of this team, and I’m probably more sad than frustrated that the players weren't rewarded for their efforts in the end," Parker said.
Brentford seemed to be cruising to victory after taking an early lead through Damsgaard, before Igor Thiago and Kevin Schade scored to put them 3-0 ahead in just 34 minutes.
But an own goal by Michael Kayode just before half-time opened a window for Burnley, and Hannibal Mejbri set up two second-half strikes by Jaidon Anthony and Zian Flemming to make it 3-3.
Flemming found the net again, only for the goal to be overturned by VAR for an offside. Damsgaard's winner sealed the visitors' first win in three league matches, as Ashley Barnes' late equaliser was also overturned due to handball.
"We changed things tactically, and it worked, but were cruelly denied on a couple of occasions, and we didn’t get what we deserved from the game," said former England midfielder Parker, who led Burnley to the English top-flight in his first season in charge.
"It takes a certain type of player to come out and show the spirit and stand up to the challenge and the atmosphere that we had in front of us at half-time."
Burnley, who earned a promotion to the Premier League in 2022/23 only to be relegated to the Championship the following season, visit Everton on Tuesday.