Tottenham bid £14million for Zaha last summer and the winger remains on Mauricio Pochettino's radar.
Zaha, whose Palace contract runs until 2020, starred as the Eagles beat Premier League-leaders Chelsea 2-1 on Saturday, scoring one and setting up another.
The 24-year-old has six goals and seven assists in 26 league appearances this term and the number of admiring glances from bigger clubs is only likely to increase if the star maintains his current form.
Asked if he was going to find 'the enormous pots of money' needed to fend off interest and keep Zaha at the club next season, Parish told BBC Radio 5 Live's Sportsweek programme: "Enormous? Are you his agent?
"We would love to keep Wilfried, we will sit down and talk to him as soon as we're safe and I'm very hopeful of doing so."
Palace's win at Stamford Bridge was their fourth on the spin as Sam Allardyce looks to steer the club away from relegation trouble.
The Eagles remain four points above the drop zone ahead of Wednesday's trip to Southampton.
Parish said: "We're just hopeful we can carry on winning and look forward to next season. We did think it was this season, maybe we didn't spend quite enough in the summer, but next season - if we can [survive] - we're set for a pretty decent season.
But will Zaha be part of Palace's bid to kick on next term?
Parish added: "[Zaha] doesn't have a release clause, I'm pleased to say. We will be sitting down as soon as possible to make that happen.