Preheat the oven to 400 degrees F.
In a large skillet set over medium-high heat, heat the oil. Add the sausage and cook, breaking it up with a wooden spoon, until browned throughout, about 5 minutes. Using a slotted spoon, transfer the sausage to a paper towel-lined plate.
Add the onion and garlic to the pan, adding extra oil, if needed. Cook, stirring, for 1 minute. Add the butternut squash and parsnips and cook until the vegetables are tender, 8 to 10 minutes. If the squash and parsnips start to stick to the pan, add 1 tablespoon water. Season to taste with salt and pepper. Remove the pan from the heat and pour off any excess liquid. Stir in the reserved cooked sausage.
In a medium bowl, whisk together the ricotta, eggs, spinach, 1 teaspoon salt, and 1/2 teaspoon pepper.
In a separate bowl, combine the mozzarella, Parmesan, and basil.
Spread one-third of the tomato sauce in the bottom of a 9-by-13-inch baking dish. Arrange a layer of lasagna noodles on top. Evenly distribute one-third of the ricotta mixture over the noodles, followed by one-third of the vegetable-sausage mixture. Then sprinkle with one-third of the shredded cheese mixture. Repeat to make two more layers, ending with the shredded cheese mixture on top. (The assembled lasagna can be kept tightly covered in foil in the refrigerator up to 1 day before baking.)
Cover the dish with foil and bake for 30 minutes. Remove the foil and bake until the top is golden brown, about 15 minutes. Let cool for at least 10 minutes before serving.