There were no goals after 90 uninspiring minutes, but the Bantams got the better of their League One opponents 3-2 from 12 yards. The decisive moment came when Pools' Evan Horwood's kick cannoned off goalkeeper Jon McLaughlin, hit the bar, bounced on the line and rolled to safety.
Parkinson's side scored three from the spot through Nathan Doyle, Nahki Wells and Alan Connell, while Andrew Davies and Garry Thompson missed.
He said: "It's pleasing to get through and I think we deserved it. We had some good chances at a League One club, but we defended well and that's what pleased me.
"When Hartlepool pressed we looked solid at the back and that pleased me more than anything. The back four gave us a good chance to go forward. We stuck to the job, the shape was good. We have players coming back but some lads have been given a chance and they took their chance.
"I watched the penalties and felt confident. Jon is sharp around his goal and we had a lot of strikers on the pitch. The decisive penalty at the end was a strange one, the linesman didn't do anything and no-one realised what happened."