The Bantams were already three goals down when goalkeeper Matt Duke brought Jonathan de Guzman down in the box less than an hour into the contest. Referee Kevin Friend brandished the red card and, when De Guzman netted the spot-kick, Bradford were four down but they survived further punishment until stoppage time, when De Guzman netted his second.
"It was damage limitation after the red card," said Parkinson. "We had to make sure they didn't score seven or eight, which could have happened quite easily. The players deserve great credit to go 30 minutes with 10 men against a team of that quality and only concede right at the death."
After such a thrilling run to the final, during which they accounted for three Premier League scalps, plus Championship promotion contenders Watford, Bradford found Swansea far too good.
Parkinson was acutely aware the game was up long before Duke's dismissal, which will keep him out of Wednesday's League Two clash with Dagenham and Redbridge.
"The referee could have used his common sense in the context of the game," he said. "We were 3-0 down against a Premier League team and a penalty had just been awarded against us. I don't think we were going to come back to win 5-4.
"He could have used a bit of discretion and just give Matt a yellow card and that would have sufficed."
Bradford remained in the south on Sunday night to overcome their exertions, with Parkinson insisting his players have nothing to reproach themselves for.
"I would have loved to make more of a game of it but it was a tough afternoon," he said.
"One of the Swansea players' wages probably covers our budget for the whole season. That is the gulf we were dealing with."
- Sports & Recreation
- Jonathan de Guzman