Spurs go into Thursday night's quarter-final first leg against Basel as joint-favourites for a competition which helped make Andre Villas-Boas' name as a manager.
Villas-Boas won the Europa League with Porto and has consistently fielded strong teams during his first season at White Hart Lane, a policy which has left the Londoners within sight of a first trophy since 2008.
Parker, whose only silverware as a top-flight player was an Intertoto Cup winners' medal in 2006, said: "It would be massive for me personally and I think it would be massive for a lot of us in the changing room and for the club as well.
"As well as us as a squad, when I say that I obviously mean the fans as well. I can't remember the last time we won a trophy.
"For the fans, it would be something very special - it would be something that shows the club is moving in the right direction and would be pleasing.
"We have gone on a very good run and it is a tough competition with good teams in it.
"We wouldn't want to look too far ahead because obviously we have got two difficult games coming up, but you're at the business end now so there is a chance you can go all the way."
Spurs remain on course to qualify for next season's UEFA Champions League, and Parker said: "This is the business end of the season.
"This is where it defines you. We have worked hard to get to this stage and are in the quarter-finals and are sitting healthy in the league.
"I suppose this is the most important point now. You need to maintain and keep picking up results and finish and maintain your objectives and get there at the end of the season."