As a League Two outfit competing on the grandest of stages, few gave the Bantams much of a chance prior to kick-off - even after seeing off Wigan Athletic, Arsenal and Aston Villa in earlier rounds.
Bradford never got going against Swansea, falling behind inside 16 minutes and going on to concede four more.
Parkinson's men also had goalkeeper Matt Duke sent off early in the second half, with the fourth-tier hopefuls unable to replicate the heroics which had carried them to Wembley.
"We would have liked to make more of a game of it," Parkinson told Sky Sports.
"Conceding the goals when we did made it really difficult for ourselves. Second half, we changed it at half-time but then conceded a goal straight away, which was another setback.
"To then get our goalie sent off, I thought was a harsh decision. In the context of the game, the ref could have just looked at that and said we're 3-0 down with a penalty against us, a yellow card would have sufficed. That was then game over.
"Swansea are a very good side. What they have done to us today they have done to Premiership teams. We are many places below them in the Football League and it was tough for us today. But enormous credit to our boys for getting to the final, it's an incredible achievement.
"We would love to have given our supporters something to cheer today and been in the game longer, but we were unable to do it. They are a good side and things didn't quite go our way.
"We came here today to do what we have done against other Premier League opposition, which was sit back and try to hit them on the counter attack, but Swansea are an exceptional side."
Parkinson's contract at Bradford is due to expire in the summer, and he has been linked with various other posts on the back of a remarkable cup run but he insists he would welcome the opportunity to prolong his stay at Valley Parade.
He added: "Myself and all the staff have enjoyed our time here and would love to stay."