Palmyra Gym Directory. Click here for a comprehensive list of gyms, health clubs and fitness centres in Palmyra Western Australia. Looking for a suitable gym, health club, fitness centre, recreation centre or aquatic centre close to home or work? Click here for Australia's leading gym directory, the Good Gym Guide. Palmyra Gym Directory. Click here for your free 14 day pass. Health Clubs Palmyra Western Australia.