Preheat oven to 375 °F (190 °C). Cook lasagna noodles according to package directions; drain and set aside.

Heat olive oil in a skillet over medium heat. Add mushrooms and sauté until golden, about 6–8 minutes. Add garlic and cook for 1 minute more. Stir in spinach and cook until wilted. Season with salt and pepper. Remove from heat.

In a medium saucepan, melt butter over medium heat. Whisk in flour and cook for 1 minute to remove raw flour taste. Gradually whisk in milk until mixture is smooth and thickened, about 3–4 minutes. Add cream cheese and stir until melted. Season with salt, pepper, and nutmeg if using.

Spread a thin layer of sauce in the bottom of a 9×13-inch baking dish. Layer 3 noodles on top of the sauce. Spoon half the mushroom-spinach mixture over the noodles, then pour about one-third of the sauce. Sprinkle with one-third of the mozzarella and a bit of Parmesan.

Repeat layering: 3 noodles, remaining mushroom-spinach, another third of the sauce, and another third of the cheeses. Top with final 3 noodles, the remaining sauce, and finish with the rest of the mozzarella and Parmesan.

Cover with foil and bake for 20 minutes. Remove foil and bake for an additional 10–12 minutes, until the top is bubbly and golden.

Let rest for 5–10 minutes before slicing. Garnish with chopped parsley.