Parkinson's footballers have already eliminated three top-flight teams on the road to Wembley and the minnows are confident of adding Swansea to an impressive list of scalps that includes Arsenal, Aston Villa and Wigan.
Swansea is a firm favourite to win the trophy, but former Charlton boss Parkinson sees no reason why his collection of unheralded journeymen can't upset the form book one more time.
With Bradford currently 11th in English football's fourth tier, Parkinson's team has only slender hopes of a promotion challenge, so the City boss is happy to admit he is focusing all his attention of winning at Wembley, which would bring not only the club's first major silverware since the 1911 FA Cup but also a place in next season's Europa League.
"All along the league has been our priority. I cannot hide away from that," he said. "But we are in the final at Wembley and have a chance of getting into Europe.
"So I can honestly say now, if we had the choice of two things, it would have to be winning the cup.
"We are a League Two team. We have created history by getting this far. It would be truly amazing if we could pull it off."
While Parkinson would love to lift the trophy this weekend, he acknowledges that Bradford's long-term stability and growth is more important after years of financial problems after the club's relegation from the Premier League in 2001.
"The chairman has done a great job clearing the debts but now it is their job - and mine - to ensure this extra revenue is used to keep improving the structure of the club and make sure our supporters have a club that will flourish in years to come," Parkinson added.
"As much as we want to enjoy this weekend, it is really important this revenue is used wisely. I am sure it will be."