Brentford produced a stirring second-half display to maintain the pressure on the leaders with Clayton Donaldson breaking the deadlock with a sizzling low drive from the edge of the box.
By the time George Saville's bobbling shot crept inside the upright in the 73rd minute, Bradford's resolve was broken.
Wayward finishing looked like sabotaging the Bees' bid to stay on the coat-tails of London rivals Orient, who could only manage a draw at Notts County.
Brentford dominated possession for long spells, but the Bantams' well-organised lines of defence ensured the scores stayed level.
"If we'd held it for another 10 minutes I think we would have got something from the game as the crowd were starting to get a bit edgy," said Parkinson.
"I was really pleased with the effort and organisation we showed until the goal, but we talked about what we needed to do and when we didn't do it, we paid the price."
He added: "We came here without several key players and James Hanson in particular was a huge loss, but he had a slight groin twinge and we couldn't risk him.
"We had done a lot of work on the system we wanted to use with him at the top, so it was a loss but I was pleased with our resilience and really couldn't have asked for any more from the lads."
After the opener, Brentford really should have punished the visitors, but squandered a string of chances to make the scoreline more impressive.
Donaldson headed Adam Forshaw's free-kick just past the post and should have made it a brace when he somehow skied Marcello Trotta's near-post cross over the bar from close range.
Jonathan Douglas headed against the bar and Saville forced a point-blank save from City goalkeeper Jon McLaughlin as Brentford piled on the second-half pressure.
City's only real effort of note came just before the break when Adam Reach beat three defenders only to fire over the bar.