The Eagles knew it was going to be tough to make the step up into the top tier, with a limited budget at Selhurst Park leaving them a long way behind their rivals.
Palace have, however, done all they can to give Holloway the best possible tools for the job.
A difficult opening to the season has brought five defeats in six and just three points, but co-chairman Parish is confident the tide will turn.
He said in the South London Press: "I believe in the manager and I believe in his ability to create a winning team.
"We need to keep the ball better. You can't keep giving it away in this league because, unlike in the Championship, you won't get it back. Ian knows that and it was what we were trying to work on over the summer transfer window, bringing people in to try and help us do that.
"I've got every faith in Ian to sort it out. There are a lot of new players and we're trying to find both a formation and a way of playing.
"We didn't get what we deserved in the first few games and that puts us where we are. But we've got to keep believing. I know we're good enough to stay in this league. We just need to find a bit more of a goal threat and believe.
"There is a lot to be positive about. We've just got to pull together and get some results."
Parish has also rubbished reports suggesting that Palace officials have held discussions with a Dubai-based consortium regarding a possible takeover.
He added: "There is no truth in it at all. I've no idea where that came from. I haven't met anybody. If we end up selling to a Dubai-based consortium then I'll be called a liar.
"All we are looking at is how to improve on a poor run of results."