The club have moved quickly to secure the services of the playmaker, who scored eight goals in 26 appearances for Albion last season.
He previously spent time at Silsden and Harrogate Railway, playing under Mick O'Connell at all three clubs.
Discussing the move, Crosby said: “I'm delighted to have signed for BPA.
"I’m looking forward to getting started for the new season to continue to challenge myself and push for promotion with the team."
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