Spurs reached the semi-finals of the competition last season but were thrashed by London rivals Chelsea.
Parker believes that Tottenham are good enough to do better under Andre Villas-Boas and lift the trophy for the first time since 1991.
He told The Sun: "It's quite some time since Spurs won it. Last year was close and hopefully if you get good draws, you can go some way. A club like ours, at our sort of level, we can definitely win the Cup this year.
"I read that the final is going back to the last game of the season, which is what you remember as a kid, Wembley and stuff, for me and a lot of the lads it's vitally important. We got to the semis of the FA Cup last year but, other than that, I haven't had a lot of joy.
"Last year was massive for us, It was very disappointing in the semi-final when we were beaten quite heavily by Chelsea. Last year was the closest I've come. There's not a day goes by when I don't think, 'If I could get a medal, or win something big'. And the FA Cup is something big."
Villas-Boas's side play Leeds United later today in a fourth-round tie at Elland Road.