Winger Young was booked for tumbling over after a challenge by Palace's Kagisho Dikgacoi in United's 2-0 win at Old Trafford.
Later in the first half, the referee awarded the home side a penalty and sent off Dikgacoi after the pair clashed again, although television replays suggested Young had tumbled easily under minimal challenge.
"The only player in the incidents that was honest was Kagisho Dikgacoi and he's sent off and banned for the next match," Palace chairman Steve Parish told the BBC on Sunday.
"Ashley Young has a yellow card and three points and we have no points and one less player to pick from for the next game.
"If preventing a goal-scoring opportunity is a straight red then trying to create one by cheating should be a straight red also. It might have cost us a point that might keep us up.
"(We) need to get some momentum behind a straight red for a dive."
Young has gained an unenviable reputation as a player prone to falling over in the penalty area and the latest incident has led to calls for him to be banned.
Former Premier League referee Graham Poll, writing in his column in the Daily Mail, accused Young of cheating.
"He should be banned for five games, then he might stop diving to try and con referees."
Former United manager Alex Ferguson said he had "had a word" with Young last year after several high-profile incidents in which the former Aston Villa player had fallen over.
Current manager Moyes took Phil Neville to task when he was in charge of Everton last season after the midfielder dived in the Merseyside derby and looks likely to continue his stance at United.
"I don't want my players diving. It's not what I want. Dikgacoi definitely throws his leg out but Ashley put his leg into his leg," Moyes said of Saturday's controversy.
"I don't like the rule where every time it is the last man it means it is (a red card). I thought it was harsh."
Last season in the Premier League, 34 yellow cards were shown for diving with former Tottenham Hotspur forward Gareth Bale the worst culprit, picking up six of them.
The Football Association can take retrospective action against players they feel were guilty of simulation, even if the referee deemed nothing was wrong at the time.