Pulis guided Palace to Premier League safety after taking over from Ian Holloway last November, and was recognised for his efforts with the division's manager of the year award.
However, reports suggest the Welshman has clashed with Parish over a failure to land a host of close-season transfer targets.
And as the odds tumbled for Pulis to become the first managerial casualty of the season, Parish was unable to quell reports of a disagreement with the former Stoke boss.
"I don't know, I'm doing something completely different," Parish is quoted as saying when contacted by the Evening Standard. "I can't comment. I don't know anything about it."
Asked if he could guarantee Pulis would be in charge for Saturday's Premier League opener at Arsenal, Parish added: "I really don’t know anything about it."
Palace have added Fraizer Campbell, Brede Hangeland and Chris Kettings to their squad since the end of last season, while a reported deal for Liverpool defender Martin Kelly has yet to be finalised.
