In a deep skillet over medium high heat, heat the olive oil. Add the onions, zucchini, squash, mushrooms, and ½ teaspoon salt. Cook, stirring occasionally, for 5 minutes, until the vegetables are beginning to soften. Stir in the garlic and cook another 30 seconds, until fragrant.
Add the tomatoes, tomato paste, oregano, the remaining ½ teaspoon salt, and pepper. Increase the heat to high to bring the mixture to a rapid simmer, then reduce the heat to medium.
Make 4 to 5 wells and spoon in the ricotta cheese. Break the lasagna noodles in half and lay on top, overlapping as needed. and spooning some of the liquid on top. Cover the noodles with the mozzarella cheese slices.
Cover the skillet with a lid and cook for 15 minutes, until the noodles are tender and the cheese is melted.
Sprinkle with Parmesan cheese and chopped fresh Italian parsley and serve directly from the skillet.
Do not cook the noodles ahead of time, and be sure to use the no-cook/oven-ready lasagna noodles.