The Clarets fought back from 3-0 down to draw 3-3, before Mikkel Damsgaard struck on 93 minutes for the Bees' win.
Parker said afterwards: "We can be critical of the first half, of course we can. We gave ourselves a mountain to climb.
"We looked soft in the first half - we lost duels and we were second best for large parts in a difficult environment. I could see the hurt in the players eyes at half time with what they went through in the first half.
"But our reaction in the second half - the fight and the spirit - was incredible."
On the fight-back, Parker also said: "We were incredible in the second half.
"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. 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.
"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."