Heat oven to 450° F with the bottom rack set in the lowest position. Toss the mushrooms with 2 tablespoons of the oil, ½ teaspoon salt, and ¼ teaspoon pepper on a large rimmed baking sheet and spread in an even layer. Roast, stirring once, until tender, 12 to 15 minutes.
Lightly oil another large rimmed baking sheet. Stretch the dough into a large oval and place on the prepared sheet. Brush with 1 tablespoon of the remaining oil. Scatter the roasted mushrooms, cheese, and thyme evenly over the dough. Bake until the mushrooms and crust are well browned, 14 to 16 minutes.
Meanwhile, whisk together the lemon juice, chili, and the remaining 2 tablespoons of oil in a bowl. Drizzle on the pizza and top with the parsley.