The 32-year old, who joined Martin Jol's side from Tottenham, spent 18 months at Fulham's West London rivals, although he only managed 28 appearances in all competitions.
Parker turned down the chance to work with former boss Harry Redknapp at QPR to make the move and he says he wants to make an impression upon the Fulham faithful.
"I just want them to respect me. They've seen the way I play and the character I am," Parker said.
"Every club I've been at I've forged a relationship with the fans. I hope it'll be the same here."
The midfielder also added that a move to the Championship was not something which interested him, admitting that he was pleased to remain in England's top-flight.
"It dragged on and when Tottenham was somewhere I was no longer going to stay, there was a lot of speculation. I'm delighted to be here. Staying in the Premier League was important," he said.
"Fulham is a club I've always enjoyed playing. I'd like to see myself here for some time."