Preheat oven to 425°F (220°C). Rub squash halves with 1 tbsp olive oil, season with salt and pepper, and place cut-side down on a baking sheet.
Pat chicken dry and toss with remaining olive oil, lemon zest, juice, garlic, oregano, thyme, salt, and pepper.
Arrange chicken on the sheet around squash; add 1/4 cup broth to the pan if you want extra pan juices.
Roast 35–45 minutes until chicken reaches 165°F and squash is tender. Broil 2–3 minutes if you want crisp skin.
Let squash cool 5 minutes, shred flesh with a fork into strands, and toss with pan juices and parsley. Serve chicken on top or alongside.