Place the diced onion and minced garlic in the bottom of the slow cooker. Add the chicken breasts on top.
Pour in the drained cannellini beans and chicken broth. Sprinkle the oregano, salt, and black pepper over everything.
Cover and cook on low for 6 to 7 hours, until the chicken reaches an internal temperature of 165°F and shreds easily.
Remove the chicken to a cutting board and shred it with two forks. Return the shredded chicken to the slow cooker.
Stir in the lemon zest and lemon juice. Let the soup sit for 5 minutes so the flavors blend.
Ladle into bowls and top each serving with the chopped fresh parsley.