Heat oil in a large skillet over medium. Add onion and sauté until soft, about 3 minutes.
Add minced chicken, breaking it up with a spoon; cook until no longer pink, 5–6 minutes. Season with salt, pepper, and oregano.
Stir in garlic and orzo; toast 1 minute. Add broth, lemon zest, and lemon juice. Bring to a simmer.
Cover and cook 8–10 minutes, stirring once, until orzo is tender and most liquid is absorbed.
Fold in spinach and Parmesan, if using, until wilted. Adjust seasoning and sprinkle parsley before serving.