Heat the olive oil in a large skillet over medium heat until shimmering.
Add the sliced mushrooms, sliced zucchini, minced garlic, and dried thyme to the skillet and cook, stirring often, until the vegetables are softened, about 5 minutes.
Push the vegetables slightly aside and add the uncooked whole-wheat spaghetti to the pan; pour in 4 cups of water and season with kosher salt and ground black pepper.
Bring the mixture to a boil, then reduce heat and simmer, stirring occasionally, until the pasta is tender and most of the water is absorbed, about 10 minutes.
Remove from heat, stir in the grated Parmesan until melted and evenly coated, then sprinkle with chopped parsley and serve.