Zaha, 24, has been in excellent form in the Premier League this season, scoring six goals in 28 appearances for Palace. He has more than three years left on his contract with the club.
Spurs had a £12m ($15m) offer rejected for the former Manchester United winger last summer and Palace also reportedly turned down fresh approaches from the north London club for the player in January.
Parish insisted that Zaha was a key member of the Palace squad and that it would be very difficult for another club to prise him away from Selhurst Park in the summer.
"It will be really hard for him to go in the summer, really difficult," he told Sky Sports.
"We're confident [he will stay]. Wilfred is focused on the end of the season and keeping us in the division. He's been here since he was very young. We'll sit down at the end of the season and I'm very confident we can keep him.
"We want to go up the league. We thought the season would have gone better than it has turned out but we still have the chance to finish relatively high. I think we're only three or four points off 10th.
"We're looking to improve the squad and improve the squad and Wilfried is an intrinsic part of what we're doing."
Zaha made two friendly appearances for England but switched his international allegiance to Ivory Coast last year and competed at the African Cup of Nations in January.
Palace boss Sam Allardyce said earlier this month that he would have tried to convince Zaha to play for England if he was still the Three Lions manager.
"As far as I know, and I'm not apportioning blame, it wasn't done when it could have been done - so Wilf made a decision to play for the country he was born in," he was quoted as saying by London Evening Standard.
"It's happened, and certainly at the moment he is playing outstanding football for us and in the Premier League: the hardest place to perform consistently as it's such a competitive league."