Place the chicken breasts in the slow cooker. Add the diced carrots, celery, onion, and minced garlic around the chicken.
Pour in the chicken broth. Add the lemon zest, dried oregano, bay leaf, salt, and black pepper. Stir gently to combine.
Cover and cook on low for 6 hours or on high for 3 hours until the chicken is fully cooked and tender.
Remove the chicken from the slow cooker and shred it with two forks. Return the shredded chicken to the pot.
Stir in the orzo and lemon juice. Cover and cook on high for 20 minutes until the orzo is tender.
Add the fresh spinach and stir until it wilts. Remove the bay leaf before serving.
Ladle the soup into bowls and sprinkle with chopped fresh parsley.