Heat oven to 350° F. Oil an 8-inch square baking dish. Heat 3 tablespoons of the oil in a large skillet over medium-high heat. Add the onion and cook, stirring, until tender, 3 to 4 minutes. Add the mushrooms and cook, tossing occasionally, until tender and any liquid has cooked off, 4 to 6 minutes more. Remove from heat and mix in the parsley.
Meanwhile, in a bowl, whisk together the eggs, milk, ¾ teaspoon salt, and ¼ teaspoon pepper. Add the bread, cheese, and mushroom mixture and toss; transfer to the prepared dish. Bake until golden brown and set, 50 to 60 minutes.
In a bowl, toss the greens with the vinegar, the remaining oil, and ¼ teaspoon each salt and pepper. Serve with the strata.