Gary Hooper's deflected effort was cancelled out by Pajtim Kasami, and, with both side's pushing for a winner, it was the Cottagers' veteran midfielder who found a breakthrough
Gary Hooper opened the scoring with a long-range strike that deflected off Aaron Hughes and looped over David Stockdale, but Rene Meulensteen's men hit back though Pajtim Kasami, whose low free kick flew under the wall and bounced into the back of the net.
Sascha Riether was forced to clear Leroy Fer's header off the line just after the break and Fulham would capitalise on that let-off late on, with Parker firing a powerful effort towards goal that rippled the back of the net.
The hosts were held to a scoreless draw at Sunderland on Saturday and manager Chris Hughton named the same team for the visit of Fulham's, while Maarten Stekelenburg's absence with injury was the most notable of Meulensteen’s five alterations.
Aside from a long-range Kasami shot, the opening stages were tame, but the match livened 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 Riether cleared 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.
The Canaries keeper, though, was powerless to prevent Fulham scoring a late winner as Parker took a touch and blasted a shot into the net from outside the box to claim all three points.