The Bantams needed a penalty shootout to see off their opponents after the tie ended 3-3 at the end of extra time.
Ishmel Demontagnac's 43rd-minute penalty for Northampton cancelled out Will Atkinson's opener before Nahki Wells' last-minute penalty appeared to have won it for Bradford, only for Clive Platt to send the tie to extra-time with a goal in the dying seconds.
Kelvin Langmead appeared to have won it for Northampton when he made it 3-2 in the 109th minute, but Bradford defender Carl McHugh equalised in the last minute of extra time and Parkinson's side held their nerve to win the shootout 4-2.
"We should have won the match in 90 minutes," said Parkinson. "The game should have been done and dusted, but the lads never know when they are beaten, Carl McHugh has scored the equaliser and the rest is history."
Parkinson said he was disappointed that Bradford were not awarded a penalty when teenage substitute Adam Baker appeared to be brought down by Northampton left-back Joe Widdowson two minutes after they had gone in front for the first time.
"It should have been a red card and a penalty," he said. "Northampton then got a contentious penalty for an equaliser and that changed the course of the game."
