Preheat oven to 400°F. Rub cut sides of squash with 1 tablespoon olive oil and season with 1/4 teaspoon salt and pepper. Place cut-side down on a rimmed baking sheet. Roast until a knife slides easily through the skin, 40 to 50 minutes. Flip cut-side up and let cool slightly.

While squash roasts, season chicken with garlic powder and remaining 1/4 teaspoon salt. Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add chicken in a single layer and cook undisturbed until golden brown, 4 to 5 minutes. Flip and cook until cooked through, 3 to 4 minutes more. Transfer to a plate.

In the same skillet, melt butter over medium heat. Add garlic and cook until fragrant, about 30 seconds. Pour in heavy cream and bring to a gentle simmer. Cook until slightly thickened, 3 to 4 minutes, stirring occasionally. Remove from heat and whisk in Parmesan and nutmeg until smooth.

Use two forks to shred the roasted squash flesh, leaving it in the shells. Transfer shredded squash to a large bowl, leaving a 1/2-inch border to maintain structure. Add cooked chicken and pour Alfredo sauce over top. Toss gently to combine, then mound mixture back into the squash shells.

Return stuffed squash to the oven and bake until heated through and bubbling at the edges, 10 to 12 minutes. Sprinkle with additional Parmesan and fresh parsley before serving.