The Reds, owned by Hollywood stars Ryan Reynolds and Rob Mac, have secured a record-breaking three successive promotions and are currently in the Championship play-off places.
Wrexham are underdogs against Chelsea, the club world champions who sit fifth in the top-flight, but Parkinson says his players won't lack belief that they can spring a surprise and progress to the quarter-finals for the fourth time in the club's history.
"You've got to go into every game believing and if not, we might as well not turn up," said the Reds' boss.
"We've got to go into it with a belief but we've also got to go into it with an understanding of the underdog mentality and what boxes we've got to tick to make the evening as difficult as possible for Chelsea.
"It's always good to be here in front of our home fans.
"Saturday is a bit of a celebration, I feel, for the journey the club are on, and I want the supporters to come in and enjoy watching us play against top-class opposition.
"It's been a long time since we've been in this round, we've got a rich history in the FA Cup and it's important on Saturday evening our supporters and the wider audience that are going away from the game have seen a team which has respected that."
Parkinson was at Villa Park to see Chelsea beat Aston Villa 4-1 on Wednesday night.
Manager Liam Rosenior is expected to make changes against Wrexham, with the Blues facing Paris Saint Germain in the Champions League next week, but the Reds' boss said it was important to see tomorrow's opponents at first hand.
"It was a great game to watch. It was very entertaining and Chelsea were good on the night," he said.
"It was great to be at the game and to watch them on the bigger picture rather than on the laptop and get a good feel for what they're all about, and it's going to be a great game.
"You've got to look at the opposition but you're always concentrating on picking our team and getting us to function to the best of our ability."
Wrexham continue the promotion push with a home game against play-off rivals Hull City on Tuesday before facing Swansea City at The Racecourse in the Welsh derby next Friday and Parkinson says it's a fantastic time for the club.
"It's a great week for us" he added. "These are the weeks that you really want to be involved in as a manager, player, supporter.
"Chelsea at home in the FA Cup and then we roll into two really exciting games.
"When people moan about fixtures, we've worked hard to get to this stage of the FA Cup, we've worked exceptionally hard to be in the top six and I want us to make the most of this week, starting with the weekend."