Heat the olive oil in a large pot over medium heat. Add the chicken pieces and cook until lightly browned on all sides, about 5 minutes.
Add the onion, carrots, and celery to the pot. Cook for 4 to 5 minutes until the vegetables begin to soften.
Stir in the garlic and cook for 1 minute until fragrant.
Pour in the chicken broth and bring the mixture to a boil. Add the orzo, salt, and pepper. Reduce heat to a simmer and cook for 8 to 10 minutes, stirring occasionally, until the orzo is tender.
Stir in the lemon zest, lemon juice, and spinach. Cook for 2 more minutes until the spinach wilts. Remove from heat and sprinkle with parsley before serving.