Preheat oven to 400°F. Brush squash with oil, place cut-side down on a baking sheet, and roast 30–40 minutes until tender.
While squash roasts, heat oil in a large skillet over medium-high. Season chicken with salt and pepper and cook until browned and cooked through, 5–7 minutes; remove and set aside.
Add butter to skillet, sauté onion until translucent, then add mushrooms and cook until browned, about 5 minutes. Stir in garlic and cook 30 seconds.
Pour in Marsala to deglaze, scraping up browned bits, then add chicken broth and simmer 3–4 minutes to reduce slightly.
Stir in cream and return chicken to pan; simmer until sauce thickens, 2–3 minutes. Adjust seasoning.
Shred squash strands with a fork, divide among plates, and spoon chicken and sauce over top. Sprinkle parsley if using.