This is a triumph of piecing together back streets to provide an alternative to busy roads. In particular, it offers a much more pleasant alternative to Mill Road.
The traffic-free Tins and Snakey Path take you to Brookes Road; don't cross onto Mill Road, but dogleg left then right to follow the cycle route towards Greville Road. Another dogleg takes you across the (heated!) cycle bridge crossing the railway. From here (and the station), don’t follow Hills Road into town, but rather the signposted back-street route via Devonshire Road and Gresham Road which brings you out opposite Parkers Piece.