Rene Meulensteen's side had won just one of their previous nine league outings and they looked to be heading towards a disappointing Boxing Day when Gary Hooper's deflected shot put Norwich ahead.
But Pajtim Kasami, who last scored in the memorable 4-1 victory at Crystal Palace, levelled before half-time and Parker struck a marvellous winning goal with just three minutes remaining.
Meulensteen made five changes and had the likes of Maarten Stekelenburg, Brede Hangeland, Phillippe Senderos and Dimitar Berbatov all missing through injury.
The Dutchman opted to field Adel Taarabt in attack once again but Norwich started well and will be ruing a number of missed chances that came and went before Kasami's equaliser.
Chris Hughton's side also spurned a couple of chances in the latter stages before Parker bagged just his second Fulham goal to take all of the points.
Switzerland international Kasami had the first decent chance as he found plenty of space and rifled in a shot that England goalkeeper John Ruddy had to parry in to the ground to take the sting out of the ball.
The hosts soon came back into the game and Nathan Redmond was enjoying plenty of possession in the opening exchanges. His dangerous ball across the box was begging for a touch from an on-rushing Norwich player but flashed in front of everyone.
The opening goal came after 13 minutes as Norwich were starting to assert themselves.
Michael Turner stepped out of defence and played a pass forward to Johan Elmander, who touched the ball towards Hooper.
The summer signing from Celtic took a touch before firing in a shot which took a big deflection off of Aaron Hughes and looping over David Stockdale in the Fulham goal to give Hooper his third goal in four games.
Kasami, recalled to play off Taarabt, was the main point of the visitors' attacks and he flashed another shot wide on 20 minutes as Fulham looked to get back into the game.
Norwich could have doubled their lead shortly afterwards as Elmander's pressing forced a mix-up between Stockdale and Hughes but the former Bolton man could not pick out a yellow shirt as he looked to pull the ball back.
Hooper passed up the best chance to put Norwich firmly in the driving seat.
Sascha Riether missed a chance as he headed a ball back towards Stockdale that fell straight into the path of Hooper but he saw his effort roll harmlessly wide.
For all of Fulham's defensive woes, Meulensteen's side were creating enough chances at the other end to keep Norwich on their toes.
First Kieran Richardson arrowed a free-kick inches over the crossbar before Damien Duff failed to divert Kasami's cross past Ruddy moments later.
Ruddy's clearance was charged down by Parker, the ball nestled in the open goal but the effort was chalked off as it came off the arm of the midfielder.
The equaliser did come shortly afterwards as Parker was felled by Bradley Johnson and Kasami bent a low free-kick under the jumping wall and out of reach of Ruddy.
Norwich looked brighter at the start of the second half and came close to retaking the lead when Sebastien Bassong rose to head Robert Snodgrass' corner goalwards, only to see Riether head it off the line.
The Canaries defensive line was then caught out with a long ball up field, with Ruddy making a smart save to keep out Taarabt's volley.
The home fans were up in arms on the hou -mark as they felt Elmander had been forced to ground inside the penalty area.
They were soon in better spirits as record signing Ricky van Wolfswinkel was introduced from the bench for his first appearance at Carrow Road since October 26.
It was his fellow Dutchman Leroy Fer who was next to come close to grabbing a second for Norwich, climbing to head a Snodgrass corner straight into the hands of Stockdale.
With 15 minutes remaining Fulham put together a flowing move but Steve Sidwell saw his shot blocked bravely by Bassong.
The visitors' next chance was more befitting of the game as a goalmouth scramble at Ruddy's back post saw substitute Hugo Rodallega come close to bundling the ball over the line before the Norwich stopper gained control of the situation.
But it was another rare moment of class that decided the game as Parker ran on to a loose ball and hammered a drive past the despairing Ruddy to give Fulham a much-needed win which keeps them inside the bottom three on goal difference alone.