Parkinson's side have already dispatched Premier League sides Wigan, Arsenal and Aston Villa on their way to Sunday's Wembley final against Swansea.
Bradford's long-term goal is to regain their status as a Premier League club but Parkinson recognises the extraordinary nature of their cup run and is happy to see their league ambitions delayed by another year.
"All along the league has been our priority. I cannot hide away from that. But we are in the final at Wembley and have a chance of getting into Europe," he said.
"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.
"Mentally we have come through some big games; Wigan, Arsenal, two (legs of the semi-final) against Aston Villa. But this is another level above that.
"The next challenge is can we handle the big one, at Wembley. I am ready for that and I am confident my players will meet it too."
If Bradford do win the final they will be presented with the trophy by Fabrice Muamba, who suffered a cardiac arrest playing for Bolton against Tottenham in March last year.
Muamba, who was subsequently forced to retire, said: "It's been such an exciting tournament this year and it will be a great honour to be presenting the trophy to the winners. I know it will be a very special atmosphere on Sunday."
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