A game of few clear-cut chances was settled in first-half injury time by Harry Arter's left-foot effort from just outside the penalty area which flew into the top corner of the net.
It was the midfielder's second goal in three matches and gave Fulham a third straight victory and moved them to within four points of second-placed West Brom, who host Derby on Wednesday.
"I am very proud of my team. It's a big result for us but there's a bigger picture," said Parker.
"We were also with Tom Cairney today and players have had to step up. Harry has done that, he was fantastic today in terms of the goal and it was a fantastic strike.
"That's two in three now for Harry. We lost Mitrovic three games ago and we have had to find another way, the players have stepped up."
Fulham stay fourth in the Championship table but have narrowed the gap, temporarily at least, on the top two and Parker said: "Of course that's where we want to be.
"It's at this stage where seasons peter away or we go for it. We move on to Friday night and Cardiff and the way we are at the moment anything can happen. Hopefully we can keep producing results."
A first defeat since the restart for Forest is likely to end their hopes of catching the top two and mean they have to settle for a play-off spot.
Reds boss Sabri Lamouchie felt both teams were well below their best.
"We have conceded two goals in the worst moments – (against) Derby in the last second and just before half time tonight. It was not a good game for both teams and a draw would have been fair," said the Frenchman
"The two teams did not play like usual. Fulham did not create a lot but they won the match, so well done for that.
"We keep moving forward now and it is on to the next game. We need the points to stay where we are in the play-offs."
Lamouchi had a complaint about the goal after the ball appeared to hit Arter's arm before he shot.
"I don't need to see a replay of the goal, it was in front of me," he said. "He touched the ball with his hand and with VAR it would be a foul, but it is not available and it is bad luck for us but we don't have time to complain. We will analyse and rest and think about the next game."