It was recently suggested that Harris, who owns two other sports franchises in the USA, was interested in assuming control of the Premier League club.
While admitting that talks are being held with "a number of people", Parish insists that there will be no imminent takeover.
"There are always people who want to buy football clubs," he told the Croydon Guardian. "Sometimes people approach you and it gets out, sometimes people approach you and it doesn't, but there is no imminent takeover. We are talking to a number of people on a number of fronts.
"I've always said that if somebody wanted to invest in the club and move it forward quicker than I or we can then we would entertain it. We look at all possibilities that are out there - and there are many - but people have been quite ahead of themselves on this one."
Palace are currently in 16th place in the Premier League table after taking nine points from their opening nine games of the season.