Bruno Guimaraes and Anthony Gordon both scored first-half goals in a 2-1 win that sent 10-man Burnley closer to the drop with a sixth successive defeat.
The Newcastle skipper opened the scoring with a spectacular opener direct from a corner, before a red-card tackle from Lucas Pires left Burnley a man down.
A baffling Lesley Ugochukwu handball in the box was penalised after a VAR review, allowing Gordon to convert the first of two VAR-awarded penalties for handball, but Zian Flemming's late spot-kick was not enough to get a point for the Clarets.
Burnley caused Newcastle early trouble from a flurry of corners, surrounding back-up goalkeeper Aaron Ramsdale on each occasion, but it was the Magpies who broke the deadlock from a set-piece when Guimaraes' audacious effort looped over his former team-mate Martin Dubravka and gave Newcastle a deserved lead after half an hour.
Newcastle dominated from then on. Gordon, who put in a superb display, threaded a ball between the Burnley defence to send Anthony Elanga through on goal, and a daft tackle from Pires gave Stuart Atwell no choice but to send the left-back off.
The half would get worse for Burnley when Ugochukwu needlessly handballed a Gordon cross with both hands in the box, forcing the referee to point to the spot after being sent to the monitor by VAR.
The second half, which featured a debut for Yoanne Wissa from the bench, saw Newcastle dominate proceedings until stoppage time, when Malick Thiaw's handball, awarded after another on-field review, allowed Flemming to convert from the spot and Burnley a glimmer of hope.
Parker's side nearly gained a miraculous point when Laurent and Flemming almost latched onto a tantalising cross with the last kick of the game.
The result sends Newcastle into the top half, whilst Burnley are left languishing in 19th place, four points from safety.
Newcastle boss Eddie Howe said after the match:
"It was a tough ending to the game, that's for sure. We concede the penalties with minimal time left. Then the whole game changes. The extra man advantage goes out the window.
"But we got over the line, won the game, and that's all that matters.
"We were sluggish starters today. We grew into it. Bruno [Guimaraes] scored a great goal. Our plan was to put the ball right on the goalkeeper today.
"There were some bright spots in our play - not enough from our perspective - we know it wasn't our best performance.
But we got a win, 10 points out of 12, we're moving in the right direction."
