Mark Yeates twice threatened for Blackpool before the break, with his second effort tipped onto the bar.
A fine run from Stephen Darby set Reid up to fire low into the bottom corner.
Blackpool's Will Aimson saw red in injury time for a second caution after a clash with James Henson.
Defeat sees Blackpool slip to within one spot of the relegation places, with the Seasiders only above Shrewsbury in 21st on goal difference.

The initiative at The Meadows, along the Melville Drive side, comes ahead of a wildlife meadow being planted.
At least 10 large tree logs have been installed by the City of Edinburgh Council after Friends of the Meadows and Bruntsfield requested them.
The logs have been donated by the forestry service.
Lesley Hinds, City of Edinburgh Council's environment convener, said: "We have been working closely with Friends of the Meadows and Bruntsfield Links over recent months to install these logs, which will not only deter drivers from parking along the edge of the Meadows, but will help our efforts to encourage and diversify wildlife in the area, along with a new wildflower meadow."
Kyel Reid scored the only goal of the game as Bradford City moved to within a point of the play-off spots with victory over relegation-threatened Blackpool at Bloomfield Road.
[NEXT_CONCEPT]
Logs have been embedded into the grass at a huge city park in Edinburgh in a bid to encourage insects and other wildlife to the area.