The Eagles looked certain to be relegated at the start of last term before the arrival of manager Tony Pulis saw them secure an 11th-place finish in the table.
"If we did the same it would be fantastic. We're not getting carried away. We know staying in the league is very difficult. Three teams go down and everybody's invested a lot of money," Parish told Sky Sports News.
"I think everybody will look at what we did last season and get a lot of belief from it. Teams we play will know a little bit more about us than they did, so if we did anything like we did last season we'll be very happy."
Palace start the new Premier League season on Saturday when they travel to London rivals Arsenal.