The tall striker seized on a loose ball after the visitors failed to clear James Meredith's cross and lashed the ball into the net with a right foot shot to crown a stirring comeback after City had gone behind in the second minute.
Tom Bradshaw scored with a close-range header for the Shrews when Ryan Woods' cross from the right was headed back into the goalmouth by Adam Reach from just beyond the far post.
Bradford suffered another blow when they lost Wells after 23 minutes with an ankle injury after a heavy challenge from defender Dave Winfield.
But they responded well before half time when skipper Mark Jones and substitute Mark Yeats had goal-bound shots blocked in a frantic spell of pressure and goalkeeper Chris Weale made a superb save from Kyel Reid.
Reid equalised in the 60th minute from Yeates' cross and the match appeared to be heading for a draw before Hanson's last-gasp winner.
Parkinson said: "Things were going against us. We lost defender Rory McArdle with illness, then Nahki Wells went off injured and we also conceded an early goal - but you have got to turn these setbacks into a win.
"No one wants to hear about hard luck stories or excuses and I was just so pleased - it was a massive win for us.
"In the second half, though, we played with calmness and composure and kept knocking on the door.
"We also proved that we can score goals without Nahki Wells. He has been a big influence for us over a period of time but this was an opportunity for others to step up and take centre stage, which they did."