Add the chicken breasts, diced onion, diced carrots, minced garlic, chicken broth, and dried oregano to the slow cooker. Season with salt and black pepper.
Cover and cook on low for 6 hours until the chicken is 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 pasta. Cover and cook for 25 to 30 minutes until the orzo is tender.
Add the fresh spinach along with the lemon zest and juice. Stir until the spinach wilts.
Ladle the soup into bowls and top each serving with chopped fresh parsley.