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Crawley's debutante central defender Claude Davis cleared off the line from Jamie Devitt before Kyel Reid gave Bradford the lead on 12 minutes with a fierce left foot shot into the bottom corner from just outside the area.
That was the first time that City had gone ahead in a match since Parkinson had taken charge, and Crawley keeper Scott Shearer prevented them from doubling their advantage by parrying a goal bound shot from Devitt.
Former Macclesfield marksman Barnett levelled nine minutes after the break by heading home Matt Tubbs' cross from the right, and midfielder Andy Drury, on loan from Ipswich, pounced to fire the hosts ahead with a well taken left-footed shot past the outstretched grasp of keeper Matt Duke.
Barnett made victory safe two minutes from time by firing in from close range having earlier missed a sitter after dispossessing Luke Oliver.
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