The Midlands 3 West South opening day encounter also saw fly-half Lewis Brightmore contribute 16 points with the boot.
Pershore took the lead early on but Park went ahead through two penalty kicks, before Green powered over to score to give Park a 13-5 advantage.
This was narrowed to a single point before half time when Pershore gained a converted try, while Park also lost another debutant (competitive game) Antony Airey to the sin-bin.
After half-time Willis opened his account for the season, Pershore responded with a try of their own as the match hung in the balance at 18-17 to the home side.
Two more place kicks from Brightmore put a converted try's difference between the teams, Pershore managed a penalty to get within striking distance, but Goode popped up with a five-pointer, improved by Brightmore to deny Pershore a losing bonus point.
Meanwhile the second team also got their season off to a flyer with a 36-15 victory over local rivals Atherstone in Warwickshire 4.
James Cole managed two tries while Dave Hickey, James "Podge" Everard, Adam "Tickle" Lawrence and Christian Smith all picked up a try each. Captain Billy McBurnie improved three of them with conversions.
Last and by no means least, Dave Paterson's third team revolution began with defeat at Shottery, in a game which saw club president John "Jub" Jenkins start on the wing, and Craig Maughan play his first game for the club.
Paterson got on the score sheet, but it was the skipper's younger brother Adam claiming the plaudits with family members watching from the sideline.
Anybody interested in playing for the thirds or second team can contact Dave Paterson on 07780332174 or Billy McBurnie on 07794677749.
This weekend coming the firsts travel to local rivals Pinley in Midlands 3 West South, while the seconds are also on their travels to tackle Bedworth Bears.
The third team have been placed in the fixture pool with the hope of gaining another fixture - watch this space.