Bring 3 cups of the water to a boil in a large saucepan over high heat. Remove the pan from the heat and stir in the oats and 1/4 teaspoon of the salt. Cover the pan and let sit for at least 4 hours or up to 12 hours.

Stir the remaining water into the oats and bring to a boil over medium-high heat. Reduce the heat to medium and cook, stirring occasionally, until the oats are softened but still retain some chew and the mixture thickens like warm pudding, about 4 to 6 minutes. Remove the pan from the heat and let sit for 5 minutes.

Heat the oil in a 12-inch nonstick skillet over medium heat until shimmering. Add the mushrooms, onion, and remaining salt and cook, stirring often, until the vegetables are softened and the mushroom liquid has evaporated, about 5 minutes.

Stir in the garlic and cook until fragrant. Add the spinach and peas and cook until the spinach is wilted and the peas are heated through, about 2 minutes.

Season the vegetable mixture with pepper to taste. Gently stir the vegetables, Parmesan, and parsley into the oats and serve while hot.