The Bantams took a 3-1 lead into the second leg of their semi-final at Villa Park, where Christian Benteke reduced Villa's deficit in the 24th minute.
Villa were well on top in the first half but James Hanson scored the vital goal on 55 minutes which meant Andreas Weimann's 89th-minute strike only succeeded in setting up a tense finish.
"Coming here two goals up we had a great chance," Parkinson said.
"In the first half Villa were excellent, but in the second I thought we were excellent and deserved it.
"We knew Villa are an attacking team that would leave space for us.
"It's dreamland. We said we had a chance to make history and it's absolutely tremendous for everyone."
The result heaped more pressure on Villa boss Paul Lambert, who was gutted at the outcome – in stark contrast to his dugout rival.
Despite knocking out Wigan, Arsenal and now Villa, Parkinson was cautious when asked about the chances of his side lifting the trophy against Swansea or European champions Chelsea at Wembley.
"I think that's going a bit far, but to get to Wembley is great for us," he added.