Both City and the Shakers missed out a tie against a top Premier League side when they were pitted against each other in a match that will be played at Bury's JD Stadium on the weekend of 9/10 January 2016.
Parkinson, who spent the four years of his professional playing career at Bury, reckons the tie has all the makings of an entertaining affair.
Parkinson said: "The tie might not quite be the money-spinning match that both sides probably wanted but we have to look at it as a chance to make it into the fourth round.
"It is a local game for us and there should be a good atmosphere there.
"I know from playing there myself that their ground can create a fair bit of noise when there's a decent crowd inside.
"We should bring a good following to the game and I'm sure it will have the feeling of a great cup tie.
"They are a good side and we had a very challenging game against them earlier this season."