After beating Wigan, Arsenal and Aston Villa en route to the final, the Bantams were looking to secure another Barclays Premier League scalp at Wembley.
However, they were outclassed by the Swans as braces from Nathan Dyer and Jonathan de Guzman, including the penalty which resulted in goalkeeper Matt Duke's sending off, and a Michu strike helped the Welsh outfit collect the first major silverware in their history.
Parkinson said: "I would have loved to make more of a game of it but it was a tough afternoon.
"One of the Swansea players' wages probably covers our budget for the whole season. That is the gulf we were dealing with.
"The lads are a bit down in the dressing room because we are a better team than we showed but Swansea's all-round movement was very difficult for us to cope with on a big Wembley pitch."
Swansea formed a guard of honour at the bottom of the Wembley steps to applaud Bradford for their achievement in becoming the first tier four side to reach a major final at the world-famous stadium.
But Parkinson did leave with a sense of regret that referee Kevin Friend had not opted to keep Duke on the pitch rather than condemning him to become the first keeper to be sent off in a League Cup final.
"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."
- Sports & Recreation