The Bantams were kicked out of the competitions of Friday for fielding an ineligible player in their second-round tie against Brentford on November 30.
Bradford did not receive written permission for Curtis Good, on loan from Newcastle, to play in the competition by the 12pm deadline on the day of the fixture, which ended 1-1 at Valley Parade.
The tie had been due to go to a replay at Griffin Park but the FA announced they had expelled City from the competition and gave them until 5pm to launch an appeal.
Speaking on the matter for the first time following his side's 1-0 league win over Torquay, Parkinson said: "I want our supporters to know it was a very minor administrative error.
"When I brought Curtis Good and Blair Turgott - from West Ham - on loan it was with the FA Cup in mind. That was the main reason.
"The first question I asked of (Newcastle manager) Alan Pardew was could Curtis Good play in the FA Cup. He said `yes' but there was a line missing from the agreement. With everything that is going on at the club the administrative staff did not pick that up.
"If it had been Manchester United it would have been massive, but in reality we have done well in the League Cup and the Johnstone's Paint Trophy and the replay at Brentford falls between two big league games.
"If we win the appeal, bring it on. But if we don't, we take it on the chin."