With Manchester United manager David Moyes watching from the stands, Town bossed the opening exchanges but struggled to create chances.
That allowed Paul Cook's men to gradually grow in confidence and cause all sorts of bother for their hosts.
Eoin Doyle and Tendayi Darikwa were especially lively while Fleetwood centre-back Ryan Cresswell was called upon to produce some big headers to stop the visitors taking advantage.
The home side thought they were ahead just before the break when Nathan Pond tapped in at the back post but the assistant's flag was raised.
After the break Chesterfield asserted themselves and took the lead from a set piece, Ritchie Humpreys' corner flicked on at the near post for Liam Cooper to apply the final touch.
Town should have found a way back when Antoni Sarcevic tumbled under the challenge of Ian Evatt but Jeff Hughes could only blaze the resulting penalty over the crossbar.
A late rally saw Town haul themselves back into it and, with one minute left, Parkin popped up at the back post to hammer home after Hughes had flicked on Junior Brown's cross.