The 20-year-old looks set for for a move to the Premier League, with Manchester United and Arsenal the interested parties.
It was revealed on Monday that the Sir Alex Ferguson's side was in pole position to sign Zaha after tabling an offer worth up to 17 million pounds, but it is likely that he will be loaned back to the club should the deal be confirmed in January.
"Right now, he's a Crystal Palace player and he'll be a Crystal Palace player in the summer," Parish told The Independent.
He added, "That's the only thing I can tell you. He'll definitely be playing for us until the end of May whatever happens.
"I've said it countless times – there's no way the player is going to leave the club in January. That's where we are and that's where we've always been with it."
Despite previously confirming its interest in the England international, Arsene Wenger stated on Tuesday that his side was "never in for" Zaha, indicating that his likely destination will be Old Trafford.