Shrewsbury led one-nil with less than five minutes remaining before the game really exploded into life to preserve Graham Kavanagh's unbeaten record as Carlisle's rookie manager.
The Cumbrians, having won their three previous games, opened brightly with Leon McSweeney firing over from distance.
Matt Robson then squandered a decent chance to give the visitors the lead when he headed a near-post McSweeney cross just wide.
Shrewsbury gradually came more into it as the first half developed and took the lead in the 40th minute, Luke Summerfield's throughball picking out Tom Bradshaw and he rifled a first-time shot past advancing goalkeeper Mark Gillespie.
Robson and David Amoo went close to a Carlisle leveller in the second half before the Cumbrians eventually equalised in the 86th minute. Amoo lifted a close-range shot high into the net after Shrewsbury were unable to clear a cross from the left.
Carlisle then snatched the lead a minute into stoppage time when Sean O'Hanlon was allowed a free header from a corner.
But Shrewsbury lifted themselves for one final effort and Parry raced through and kept his cool to fire past Gillespie and put the seal on a dramatic finale.
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- Matt Robson