Substitute James Hanson rescued Bradford from more disappointment just three days after their Wembley defeat to Swansea in the Capital One Cup final.
Luke Howell's 69th-minute goal, which came after Sam Williams had missed a penalty, looked set to give Wayne Burnett a winning start to his spell as Dagenham's interim manager following John Still's departure to Luton.
But with four minutes left, Zavon Hines's cross from the right was helped into the goalmouth where Hanson stabbed the ball past Chris Lewington for the equaliser.
Parkin said: "The players were very disappointed after the game at Wembley. The one game where they didn't show what a good team they can be was the big one at Wembley.
"This was a difficult game, but it would have been more difficult if they hadn't put in the energy and commitment.
"I don't think there was a hangover from Wembley, but it is up to us now to move on.
"I feel there is an air of pessimism around the place and I don't know why. It's as if Bradford City are expected to win every game and the place needs a lift. Maybe there is a feeling of anti-climax.
"We kept going to the very end and that is all you can ask of your team. Give Dagenham's defence credit. Luke Wilkinson has had an outstanding season and a lot of people are looking at him.
"We made a change, bringing James Hanson and Nahki Wells on after giving them a breather from Wembley and it paid off with James's goal."
- Sports & Recreation