Zaha was the subject of a £12m bid from Tottenham Hotspur in the closing stages of the transfer window, but the bid was rejected by Palace.
Parish told talkSPORT: "The first game we had as owners, he scored the first goal and he was only 16, and his shorts were massive on him.
"He has just been an amazing part of everything we have achieved as a football club and a talisman for the fans.
"He carries the club in his heart and he cares about the football club.
"We have been lucky over the last six years to have players that care about the club and we need to maintain that.
"There are a lot of bad things said about footballers, but if there is one player who doesn't play for the money, it is Wilf. He loves playing football and there is still a lot more to come from him."