The Grecians seemingly proceeded to the second round after extra-time goals from Matt Jay and Nigel Atagana.
In this season's competition, teams can make up to five substitutions in normal time. Exeter manager Matt Taylor then made a sixth substitute at the start of extra-time, with Josh Key coming on for Sam Nombe.
Following an investigation, Football Association officials have now decided that due to the indiscretion by Exeter that the game will now be replayed at St James Park on Tuesday, November 30.
Lambasting the decision, Sparks said: “There’s no right of appeal on the decision – and we would have done so.
"I’ve been contacted by several people in the industry and the overriding opinion is they have no idea why Bradford City are returning to St James Park, let alone not playing in the second round.
"Given the nature of the incident and the effort the supporters from Bradford put in to get there and the costs we’ve already outlaid, I believe the game could and should have been played at Valley Parade.”
Adamant that the situation is an 'embarrassment' to the competition, with Bradford being the unwitting losers - with Lee Angol's dismissal in Tuesday's game still standing despite the fact that the result does not stand - Sparks continued: "It is an embarrassment to the game; an embarrassment to the FA and ultimately, we are the losers in this situation, both financially and logistically and in every sense of the world.
"They will hide behind the fact that we have been put back in a competition that potentially we would never had been knocked out of in the first place.
"The way that news was delivered to us and the time-frame we were given to give our comments in the first place; the whole thing has been treated with so little respect.
"I find it very contradictory of the FA, who are so proud of the magical competition they keep telling us about and they want us to uphold the integrity of it. I am afraid clubs like Bradford City and Exeter City are there to uphold the integrity and make it worth having.
"But when we can't even count and there's four or five human beings paid to do that on the night, it's a disgrace. It really is and I'd imagine those individuals will be back doing their job this weekend unscathed.
"Meanwhile, we are plotting another Tuesday night trip for 600 miles, which we will be paying for."