Martin Jol's side have beaten Stoke 1-0 at home and Crystal Palace 4-1 at Selhurst Park in their last two outings to move up to 14th in the table.
Two stunning goals from Pajtim Kasami and Steve Sidwell helped Fulham come from behind against Palace on Monday and they will travel to the south coast on a high.
Parker told Sky Sports: "When the results aren't going, no matter how well you are playing, you can try and gain all the positives you want from that, but ultimately if you are not winning football matches the pressure starts to mount and it was obviously a very big win for us.
"Going into the game it was pretty tense and I think it was a big game for both of us.
"Back-to-back wins is crucial and when you get back-to-back wins in the Premier League it suddenly springboards you, confidence rises and obviously you can move up the table a little bit.
"Our team is very attacking, the goals the other night and the way we won the game, we have certainly got that in our locker, but we also need to have the other side of it as well really.
"It was very pleasing and it was probably coming, but we need to maintain that now, especially this year. I think this year is a big year and we need to maintain that and stay amongst it really."
Watch Fulham take on Southampton live on Sky Sports 1 HD at 5pm on Saturday