Simmer chicken in a pot of water over medium heat to an internal temperature of 160°F. Remove from water and using two forks, pulling in opposite directions, shred the chicken. Once shredded, refrigerate to cool fully.
Once cooled, thoroughly mix the chicken, celery, onion, and mayo, in a bowl. Season with salt and pepper to taste.
Place the chicken salad on the avocado and top with cheese. Garnish with parsley.