Cook fettuccine according to package directions. Drain, reserving ½ cup pasta water, and set aside.
While pasta cooks, heat olive oil in a large skillet over medium-high heat. Season chicken with salt and pepper, then cook for 5-6 minutes until golden and cooked through. Remove from skillet and set aside.
In the same skillet, reduce heat to medium and add garlic. Sauté for 30 seconds until fragrant.
Add milk and cream cheese to the skillet, whisking constantly until cream cheese melts and sauce is smooth, about 2-3 minutes.
Stir in Parmesan cheese until melted. Add spinach and cook until wilted, about 1 minute.
Return chicken and pasta to the skillet, tossing to coat. Add reserved pasta water if needed to reach desired consistency.
Season with additional salt and pepper to taste. Garnish with fresh parsley and serve immediately.