Heat the olive oil in a large skillet or wide pot over medium heat until shimmering.
Add the sliced mushrooms, sliced zucchini, minced garlic, and dried thyme to the pan and cook, stirring often, until the vegetables begin to soften, about 5 minutes.
Add the uncooked whole-wheat spaghetti, arrange it evenly in the pan, then pour in 4 cups water and sprinkle 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.
Stir in the grated Parmesan until melted and evenly combined.
Remove from heat, sprinkle with chopped parsley, toss lightly, and serve immediately.