The 32-year-old England international completed a move to Craven Cottage from Tottenham, penning a three-year deal with Martin Jol's side.
As well as his time with Spurs, Parker has also had stints with Charlton, Chelsea and West Ham but believes he will be able to forge a bond with Fulham's fans.
"I just want them to respect me. They've seen the way I play and the character I am," he told Sky Sports.
"Every club I've been at I've forged a relationship with the fans. I hope it will be the same here.
"I'm delighted to be here. Staying in the Premier League was important. Fulham is a club I've always enjoyed playing. I'd like to see myself here for some time."