Reach gave Preston the lead when his cross from the right-hand side missed all of his teammates and ended up bouncing into the back of the net just before the 20-minute mark.
After the break, Birmingham managed to respond when Morrison latched on to the ball from Stephen Gleeson's corner and headed past the goalkeeper.
Marnick Vermijl managed to clear the ball but it had already gone over the line, resulting in an equaliser for the travelling side.
Jacques Maghoma came close to giving the Blues the lead when he headed from close range, but his effort clattered off the crossbar.
Preston, who had kept eight clean sheets in their last 10 games, are 16th in the league standings.