The 20-year-old winger is a reported target for a number of Premier League clubs but Parish has reiterated his desire to keep him at Selhurst Park for the foreseeable future.
"We've had a couple [of approaches for Zaha]," he told talkSPORT. "None that you would really take seriously. [They were] a couple of opportunistic bids from people who might not have read the papers [and known] that we're out of administration.
"What's important for us is that we keep this team together, at least this season.
"We're anxious to keep hold of him. He's on a five-year deal. We need to keep him happy. I completely understand that. We'll make sure he stays in January and we'll have another look at it in the summer."
Zaha is a product of Palace's youth academy.