In a large skillet heat 1/4 cup vegetable oil; add onions and garlic and sauté over low heat, stirring occasionally, until onions are tender and yellow. Stir in parsley, salt, pepper, and tomato sauce; simmer for about 30 minutes.
Meanwhile, peel the eggplant and cut into 1/4-inch thick slices; brush each slice with lemon juice to prevent discoloration. Beat egg slightly; whisk in milk and 1 tablespoon oil. Dust each slice of eggplant with flour, patting then shaking to remove excess flour. Dip into the egg and milk mixture; drain. In a large skillet over medium heat, heat 2 tablespoons of the oil. Fry eggplant on both sides, adding more oil as needed, until golden brown. Repeat with remaining eggplant slices, using more oil as needed. In a 2-quart baking dish, layer sauce, browned eggplant, and mozzarella, topping layers off with Parmesan cheese. Bake at 350° for 30 minutes.