Rumours suggested Parish and Holloway had fallen out as Palace looked to strengthen their squad to cope with a Premier League campaign.
Palace have been relegated in their last four top-flight seasons and Holloway regularly complained about his strength of his squad in the countdown to the current campaign.
The Selhurst Park club eventually made 15 signings, more than any other Premier League outfit, and Parish says the reports are simply not true.
Parish told the London Evening Standard: "Ian and I spoke 123 times on the last day of the transfer window.
"When reports came out that we weren't talking, that the board were unhappy and all that sort of stuff, we were in constant communication, at least 10 times a day.
"I can't give it a moment's credibility. It's ridiculous. There is just no grounds to any of it. Ian is our manager. We are very happy for Ian to be our manager.
"We all have a great working relationship and we're focused on one thing, the retention of our status in the league.
"We are very happy with the direction the club is going, very happy with the way he manages the side - the lads are all behind him at the training ground.
"I regularly go down to the training ground; there is no problem at all. Ian wants to stay here and build something."
Palace lost their first two Premier League fixtures before recording a 3-1 win over Sunderland prior to the international break.
Meanwhile, Palace defender Jack Hunt is recovering after breaking his ankle in training on Monday.
Hunt – an August signing from Huddersfield Town - is expected to miss the remainder of 2013.
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