Despite victory over high-flying Sheffield United last time out in the league, Crewe will come to Valley Parade currently bottom of the Sky Bet League 1 table.
Parkinson believes Crewe's recent win over Sheff Utd shows the quality within the visitors' playing ranks, and he knows his players will need the help of their fans come Saturday, especially with three squad members away on international duty.
Parkinson said: "It is a big day for us Saturday and for the supporters coming into the ground.
"Crewe are struggling, there is no getting away from that, but they beat Sheffield United a couple of weeks ago so it shows they can beat anyone on their day.
"That means we can't have any complacency from the players, the staff or the supporters this Saturday.
"I would like to urge our supporters coming to the game on Saturday to really get behind the team.
"We have a few players missing and this is when we really need the supporters to play their part.
"We've been on a good run of form recently so we know there is that expectancy among the fans but we certainly can't have anyone taking anything for granted.
"We know we could find ourselves in a good position in the league come Saturday night but we will need everyone giving it their all to try and help us get those three points."
City head into the Crewe fixture having collected seven points from their last three home games.
While Parkinson is happy with the Bantams' recent home form though, he did feel the need to flag up some aspects of his side's first half performance at Aldershot Town in the FA Cup last Sunday.
Parkinson added: "At Aldershot in the first half we dropped our standards and we've had a good chat with the lads about that this week.
"It is important now that we get back to how we have been playing recently.
"That means everybody doing their job for the team first and foremost.
"Following on from that, we then have to get the ball down and play with it the way we did in the second half on Sunday (vs Aldershot).
"The first half at Aldershot was disappointing and we need to roll our sleeves up and get right back on it this weekend."