Preheat oven to 400 degrees F (200 degrees C). Toss butternut squash with 2 teaspoons oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper; arrange in a single layer on a rimmed baking sheet.
Bake squash until softened and lightly browned, 15 to 20 minutes; set aside.
Meanwhile heat remaining 2 teaspoons oil in a large skillet over medium-high heat. Add chopped onion and 1/4 teaspoon salt; cook until softened and translucent, 3 to 4 minutes.
Add spinach to the skillet and cook, stirring often, until wilted, 1 to 2 minutes. Remove from heat and set aside.
Combine shredded provolone and mozzarella in a bowl. In a large bowl stir together ricotta, parsley, eggs, 1 1/2 cups of the shredded cheese mixture, remaining 3/4 teaspoon salt, and 1/2 teaspoon pepper.
Lightly coat a 13x9-inch baking dish with cooking spray. Spread 1 cup marinara on the bottom. Arrange 4 noodles side by side; spread half the ricotta mixture over noodles. Spread roasted squash evenly over ricotta and top with 1 1/2 cups marinara. Top with 4 more noodles and the remaining ricotta mixture. Spread the spinach mixture evenly over the ricotta and top with 1 1/2 cups marinara. Top with the remaining 4 noodles and 1 cup marinara.
Sprinkle with grated Parmesan and the remaining shredded cheese mixture, then cover with aluminum foil.
Bake covered at 400 degrees F (200 degrees C) for 35 minutes. Uncover and bake until golden and bubbly, about 20 minutes more. Let stand 10 minutes before cutting into 10 slices.
Tips
Roast the squash until tender and lightly browned to reduce excess moisture and deepen flavor.
Cover during the first bake to heat through, then uncover to brown and bubble the top for color.
Let lasagna rest 10 minutes after baking so slices hold together for cleaner plating and serving.