The winger, who joined United from Palace in January 2013, spent the second half of last season on loan at Cardiff City but failed to make an impact as they were relegated from the Premier League.
Under former United manager David Moyes, Zaha was rarely used at Old Trafford, but the arrival of Louis van Gaal may give him a lifeline.
However, if the Dutchman does not want the 21-year-old next season, Parish says he would be keen to take him back, providing manager Tony Pulis wanted him.
"It depends on what the new Man United manager wants but also what my manager wants," he is quoted as saying by the Croydon Advertiser.
"My manager has got to be convinced that it would be the right move for the player and the club.
"Tony has certainly shown enthusiasm during the season for that and knows what a fabulous player he is.
"But the circumstances for the player and the money and for everything would have to be right."
Parish was also quick to point out there will not be a big influx of players at the club as there was before the start of last season.
"I think we'll be looking for quite conservative options that we know can take us to the next level," he added. "We'll be trying to get value for money for the football club."