A solitary strike from Championship top goalscorer Aleksandar Mitrovic proved to be the difference at Ewood Park as Fulham earned took the three points back to the capital.
The Serbian striker struck just after the hour mark when he wriggled away from Tosin Adarabioyo before squeezing the ball into the net despite Christian Walton getting a touch on his low effort.
In a game of few chances, Rovers pressed for an equaliser, with Marek Rodak pulling off a terrific low save to stop Sam Gallagher from scoring late on.
The boss made one change to the team against the Whites, with ex-Fulham man Lewis Holtby missing out through an injury picked up against Middlesbrough a week earlier.
That allowed a return to the starting XI for midfield enforcer Bradley Johnson, who took his place in central midfield.
Stewart Downing, making his 600th career start, took his spot on the left, with Joe Rankin-Costello, who started for the first time at Ewood, lining up just behind in form frontman Adam Armstrong.
Rovers began the game well, and Sam Gallagher went close to opening the scoring after 10 minutes, but he could only float a header from a Ryan Nyambe cross onto the roof of the net.
At the other end, Walton had to be alert to tip over a headed effort from Michael Hector, who was up to connect with a corner from ex-Rovers playmaker Tom Cairney.
Walton did brilliantly soon after to tip the ball over the bar when Bobby Decordova-Reid's attempted cross took a wicked deflection off Ryan Nyambe and was heading towards the far corner of the net.
Going into the break as both teams started the game was probably the fairest outcome in a half that wasn't full of opportunities for either team.
The second half started in the same fashion, although Denis Odoi tried his luck from range only to lash the attempt high and wide.
Cairney had the next chance, but after being played through, Walton was in the way to beat away his left footed effort as the hour mark approached.
But Parker's side didn't have to wait too much longer for that elusive opener to arrive thanks to top scorer Mitrovic.
The ball broke kindly into the path of the advanced Joe Bryan, and the left back fed Mitrovic who rolled Adarabioyo and stroked the ball home, despite Walton's best attempts to keep the Serbian hitman's effort out.
Rovers responded with a double change that saw Dominic Samuel and Danny Graham enter the action, but it was Gallagher who came close to bagging the equaliser.
The ball broke the forward's way, but after unleashing an excellent strike, Marek Rodak pulled off a terrific stop down low to his right to keep his side in front.
There was still time for Walton to produce a strong save to deny substitute Stefan Johansen late on after the Norwegian came on in the dying stages of the game.
And the score stayed as it was, as Rovers suffered a first home defeat in over four months against a Fulham side chasing automatic promotion.
Don't forget, Rovers are back in action on home turf on Tuesday night when Hull City make the trip to East Lancashire, kick-off 7.45pm.
Rovers: Walton, Nyambe, Lenihan (c), Adarabioyo, Bell, Johnson (Graham, 72), Travis, Gallagher, Rankin-Costello (Samuel, 72), Downing, Armstrong (Brereton, 88).
Unused substitutes: Leutwiler, Williams, Davenport, Bennett.
Fulham: Rodak, Hector, Odoi, McDonald, Mitrovic, Cairney (c) (Johansen, 86), Ream, Decordova-Reid, Cavaleiro (Kamara, 90), Bryan, Onomah (Kongolo, 80).
Unused substitutes: Bettinelli, Christie, Knockaert, Sessegnon.
Goal: Mitrovic, 65.
Booked: Decordova-Reid, 74
Referee: Tony Harrington.
Attendance: 13,087 (781 away)