Holloway, 50, guided Palace to promotion via the playoffs last term, having joined from Blackpool in November 2012 but they have struggled to compete in the top flight and are second-bottom with only three points from eight matches.
"Ian's contract at Crystal Palace has come to an end by mutual consent," Parish told a news conference. "Ian felt that a new approach might help keep us in the division.
"I've enjoyed every minute of working with Ian. He leaves our club with his head held very high.
"I'd like to put the record straight. We've never fallen out. We've worked together brilliantly. We feel we need to move on to progress."
Holloway met Parish after Monday's 4-1 home defeat by Fulham and said the pair planned to talk on Tuesday as usual after a game but he declined to give any details.
Palace's only league win came in August against Sunderland, who were the first Premier League club this season to part with their manager when Italian Paolo Di Canio was sacked last month. Sunderland are bottom of the league with one point.
Holloway said the spirit at the club, who brought in over a dozen players in the close season, had suffered after the team's poor start.
"We need to shut up shop in this division. At the minute we've got a whole new group there," he said on Wednesday.
"Part of my talking with Steve was about am I the one with the energy left?
"I have to hold my hand up and say we didn't keep the spirit that got us up. We changed too much too quickly.
"Someone new coming in now with a tad more experience at this level will have 30 games left."
(Reporting by Josh Reich; editing by Ken Ferris)
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