Parker curled a sublime shot into the top corner beyond stranded goalkeeper Steve Collis with two minutes remaining to send the Huddersfield fans delirious.
Then the 26-year-old striker turned villain when he was sent off in stoppage time for a second bookable offence.
The Terriers broke the deadlock through Andy Butler when the defender's close-range effort was stopped on the goal-line by Danny O'Donnell, but he made no mistake from the rebound which deflected in off Crewe goalkeeper Collis.
Huddersfield scored again six minutes before the interval when Ian Craney drilled a shot past Collis. Minutes later, Crewe's Clayton Donaldson beat the offside trap and pulled a goal back.
Donaldson equalised after 58 minutes when he rolled a shot past Glennon after latching onto a pass from Byron Moore before Parker's clincher.