The defensive trio played their part last season as Palace pulled away from the Premier League drop-zone, eventually finishing in 11th place.
But they are not part of manager Tony Pulis' future plans and will be allowed to find first-team football elsewhere.
Parr and Moxey, two of Palace's long-serving performers, helped the club escape relegation from the Championship during their eventful stay and were part of the squad that secured promotion last season.
Gabbidon played a prominent role in last season's play-off final win over Watford and featured in a number of Premier League games, but fell down the pecking order after Scott Dann's January arrival.
Striker Aaron Wilbraham, another member of the play-off side at Wembley, has also been released by the club, while veteran goalkeeper Neil Alexander is to leave south London.
But the club are still holding talks with out-of-contract trio Julian Speroni, Marouane Chamakh and Kagisho Dikgacoi about fresh deals.