After last week's postponement, City and Barnsley will meet again in their second round tie to see who gets the opportunity to host Sky Bet League 2 side York City in the Area Quarter-Finals.
The Barnsley tie will be City's first JPT match at the Coral Windows Stadium since the Bantams played Sheffield Wednesday at the end of August 2011 - a game Parkinson watched from the stands after officially being announced as Bradford boss the day before.
Parkinson said: "Having a home tie waiting for the winners in the next round is certainly an extra incentive for us.
"A home tie in any cup competition is definitely an advantage and I'm sure both sides will be going all out to secure their place in the next round.
"Let's hope it is Bradford City at home to York and not Barnsley."