Gary Hooper scored for the fourth consecutive Premier League home game to give Norwich a 13th-minute lead as his shot struck Aaron Hughes and deceived David Stockdale in the Fulham goal.
The visitors generally had the better chances for the rest of the first period and, although Parker saw a goal disallowed for handball, Meulensteen's men were level in the 33rd minute thanks to Pajtim Kasami's free-kick.
Damien Duff and Hooper spurned further chances in the first period, but the match remained level, despite a tentative penalty claim from Johan Elmander.
Both teams looked set to settle for a point apiece only for Parker to fire in an unstoppable strike with less than five minutes remaining to move Fulham on to 16 points, but the London club remain inside the bottom three on goal difference. Norwich, meanwhile, are three points clear of the relegation zone.
Norwich drew 0-0 at Sunderland on Saturday and manager Chris Hughton named the same team for Fulham's visit, while Maarten Stekelenburg's absence with injury was the most notable of Fulham’s five alterations.
Aside from a long-range Kasami shot, the opening stages were tame, but the match did liven up in the 13th minute as Norwich broke the deadlock.
Hooper found himself in space just outside the Fulham area and relied on a huge slice of luck as his shot deflected off Hughes before looping over the helpless Stockdale.
Fulham had a 32nd-minute equaliser disallowed after Parker charged down John Ruddy's clearance and blocked the ball into the goal with his arm, but the visitors were celebrating for real little more than 60 seconds later.
Kasami stepped up to take a free-kick just outside the Norwich area and slotted it straight through wall before nestling in the bottom-right corner of the net.
Norwich began the second half much better than they ended the first and went agonisingly close to retaking the lead in the 53rd minute, but Sascha Riether cleared Leroy Fer's header off the line.
In the 61st minute, Elmander was incensed at the referee Jonathan Moss' decision not to award a penalty after receiving a nudge in the air by Riether.
And that proved to be the forward's last real involvement as he was replaced by fit-again Ricky van Wolfswinkel.
Hugo Rodallega almost bundled the ball over the line at the back post with 12 minutes remaining and the Colombia international was adamant a goal should have been awarded, but Ruddy appeared to keep it out.
Ruddy, though, was powerless to prevent Fulham scoring a late winner Parker took a touch and blasted a shot into the net from about 20 yards.
