Burnley came from behind twice to draw level through goals from Lyle Foster and Jaidon Anthony, but a VAR decision in injury-time handed United a penalty, which was converted by Bruno Fernandes.
Parker said: “I’m proud of this team. We struggled in the first half, we started the half slowly, but we stayed in the game.
“We made one or two tweaks at half-time - and I was immensely proud of the response. We showed heart, endeavour and quality in that second half."
Parker said of United's winner: “In the modern day and the way football is these days, it’s become very sterile.
“That’s the way the game has gone - multiple checks of every single second and phase.
“It is what it is but of course we’re hugely frustrated and disappointed with how the game has ended.”