Parish has been forced to deny the suggestions after media reports claimed that the pair's relationship had irreparably deteriorated amid disagreements around the club's recruitment policy.
Holloway has made new 10 new additions to his squad following promotion via the play-offs last May, either on permanent or loan deals, with Palace without a point after their opening two Premier League fixtures.
Former Palace boss Neil Warnock, who suggested prior to leaving his last job at Leeds United in April that it would be final managerial post, has been linked with a return to Selhurst Park, but Parish has laughed off the speculation.
"It’s such nonsense. There’s not one ounce of truth or credibility in the rumours," he told talkSPORT.
Warnock has also admitted to talkSPORT there is no truth in rumours he is poised to come out of retirement to rejoin Palace, where he spent two-and-a-half years prior to joining QPR in March 2010.
He explained: "You always get speculation about every Tom, Dick and Harry.
"I don't know where it’s come from. I don't want to get back into it at the moment. I've really enjoyed these last few months.
"I don't think I can cope with the hassle again. The phone is constantly going and you can't go on like that forever.
"Managers aren't given enough time at the moment. If you work for 12 months then you've done well."
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