Parkinson's side lost for the first time in 11 cup matches this season, but it took extra time and some of Brentford's big guns to sink Arsenal's Capital One Cup conquerors and earn a place in the third round.
Brentford twice came from behind to clinch a game that neither side really wanted. The Londoners will feel justice was done following Bradford's shock reprieve by the FA after being expelled from the competition and then reinstated.
The West Yorkshire club had fielded ineligible defender Curtis Good, on loan from Newcastle, in the initial tie, which had ended 1-1.
Parkinson, who admitted he was surprised by the FA's decision to reinstate his side, said: "We made major changes to the line-up and gave some of our returning players a chance to gain important match fitness.
"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 winger Kyel Reid, back from a long-term groin injury, put the visitors ahead in the 34th minute before on-loan Fulham striker Marcello Trotter equalised from the penalty spot for Brentford on the stroke of half-time.
The two sides were then inseparable until Alan Connell converted a 94th-minute penalty to put the visitors back in front, but the Bees struck three times in four minutes with Trotter equalising for a second time and Clayton Donaldson and Harry Forrester both on target.