It needed extra time for Brentford to sink Arsenal's League Cup conquerors and earn a place in round three. Brentford twice came from behind to eventually clinch a 4-2 win, in a replay that followed a 1-1 draw in the initial tie, after which City were expelled and then reinstated for fielding ineligible player Curtis Good.
Parkinson, who admitted he was surprised by the FA's ruling, said: "We made major changes to the side and gave some of our returning players a chance to gain important match fitness."
He added: "Obviously recent events overshadowed the game in more ways than one, but I was absolutely delighted with the performance put in by the side, especially when you consider we made nine changes to Saturday."
Bradford's Kyel Reid put them ahead on the half hour when he smashed home a loose ball from 12 yards. On-loan Fulham striker Marcello Trotta levelled just before the break, sending Jon McLaughlin the wrong way from the spot after he was fouled by City defender Carl McHugh.
Former Bee Alan Connell made it 2-1 to the cup specialists in extra time when he converted from the spot after a Shaleum Logan foul on Blair Turgott. But Trotta pounced again, firing a left-footed drive home into the top corner to equalise 10 minutes into time added on.
Substitute Clayton Donaldson made it 3-2 to the hosts, slotting home Harry Forrester's cross from a tight angle, before Forrester himself sealed it late on with a cool finish into the bottom corner. Bradford thought they had a strong shout for handball in the final minute when Leon Legge handled in the box, but their appeals were waved away.
Brentford boss Uwe Rosler, who was "perplexed" by the FA U-turn, said: "We could have done without the extra time but I was pleased with our professional approach to a game we did not expect to have to play.
"We scored some good goals and it is important to keep our run going in the run-up to Saturday's big game here against Stevenage.
"I would like to have wrapped it up in the 90 minutes but Bradford battled hard and made it difficult for us, but in the end I thought we deserved the win. We came through without any knocks and now our main priority is Saturday's game, when we will have a full house here."