Tom Mundy (40) top scored for the Shrews before Bragginton (55) and Tom Saunders (27) replied and gave Parley a sound base from which Green fired three sixes and five other boundaries in a lively 54 not out.
Parley had previously beaten Beehive & Southwick by nine wickets, Tom Saunders hitting an unbeaten 101.
Redlynch & Hale comfortably polished off a rain reduced target to beat Witchampton by eight wickets and tee up a tough looking Dorset/Wiltshire visit to Burbage & Easton Royal.
George Hayes took 4-17 and Kevin Hopkins 2-7 as Witchampton were dismissed for 124. Rain reduced the Redlynch ask to 94, a score they passed inside 21 overs with the still promising Ian Tanner making 41 and Dave Webber 29 not out.
Emsworth got a walkover and host Ifield in the West Sussex group on Sunday.