Preheat the oven to 400 degrees Fahrenheit and line a large, rimmed baking sheet with foil or parchment paper.
To prepare the squash, use a sharp chef’s knife to slice in half, vertically, from top to bottom. Use a large spoon to remove the seeds and stringy pieces.
Place the squash halves cut side up on the parchment-lined pan. Brush with 1 tablespoon of the olive oil over the squash, and sprinkle with ¼ teaspoon of the salt, then turn them over so the cut sides are against the pan. Bake until the squash flesh is easily pierced through by a fork, about 30 to 45 minutes. Leave the oven on.
Cook the quinoa according to the package instructions. Standard cooking instructions are: In a medium saucepan, combine the rinsed quinoa and water. Bring the mixture to a boil over medium-high heat, then reduce the heat as necessary to maintain a gentle simmer. Simmer, uncovered, until all of the water is absorbed, 12 to 18 minutes. Cover, and let the mixture steam for 5 minutes. Uncover and fluff the quinoa with a fork.
Meanwhile, cook the turkey in a large skillet over medium-high heat, stirring and breaking up until browned. Drain grease, if any, and set aside to cool slightly.
Add the quinoa, cranberries, green onion, parsley, garlic, lemon juice, the remaining ¼ teaspoon salt, and the remaining 1 tablespoon olive oil, along with the Parmesan and feta cheese to the pan with the turkey. Stir until the ingredients are evenly distributed. Taste and add additional salt, if necessary.
Once out of the oven, carefully turn the cooked squash halves over so the cut sides are facing up. Divide the turkey mixture evenly between the squash halves with a large spoon. Return the squash to the oven and bake for 15 to 18 minutes.
Garnish with chopped parsley, and serve warm.