Burnley run ends as Millwall snaps losing record
The decisive moment came in the 52nd minute when Millwall captain Jake Cooper powerfully headed Femi Azeez's cross beyond the grasp of James Trafford.
But the Lions, who moved up to seventh on the table with the win, were worthy winners at The Den.
Parker's side managed just one effort on target, which came in the fourth minute courtesy of Luca Koleosho, with the Clarets boss lamenting his side's display in the final third.
"We came here knowing it's a tough place to come and dominated the first 25 minutes but the game went away from us at the back end of that first-half," Parker said.
"Emotionally, with a young team like ours, we struggled in the big moments.
"We need to be more progressive in the front third but it's a really young team, a brand new team and we need to keep working hard.
"We lacked spark in the second-half and the game has been decided on a set play."
But for Millwall boss Neil Harris, he masterminded the Lions' first league triumph over Burnley since a 3-1 victory in February 2012.
It was also a third consecutive 1-0 win for Millwall. Since the start of last season, it has won 13 EFL Championship games by that scoreline, at least five more than any other side.
Harris believes his side warranted its victory, insisting his tactical decisions made the Clarets run out of ideas.
"It was a strong result and I thought Burnley ran out of ideas," Harris said.
"We grew into the game after Burnley were slightly better in the first 20 minutes, but I couldn't see them scoring against us.
"You just need a moment here at The Den and Jake Cooper produces it and, if anything, I thought we'd get the second goal.
"If we win games 1-0 it's fine and the first goal in the Championship is so important, especially for us."