The League Two outfit were the surprise packets of the competition, beating three Premier League teams on the way to the Wembley final.
Despite it coming unravelled today, Parkinson could not have been prouder of his Bantams players and the fans for the way they both played and conducted themselves throughout the league cup journey.
"We would love to have made more of a game of it. Conceding the goals when we did made it really difficult for ourselves," he told Sky Sports 1.
"But Swans are a very good side and what they have done to us today they have done to Premiership teams.
"We're 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 was an incredible achievement.
"We would have loved to have given our supporters something to cheer about, being in the game longer, but we couldn't. They're a good side and things didn't quite go our way, to put it mildly."
He added: "The highlight was our supporters, throughout the game they kept behind us and they've done the city proud, they were absolutely tremendous."