Heat the olive oil in a medium skillet over medium heat. Add the chopped onion and sauté for about 3-4 minutes until translucent.
Stir in the minced garlic and cook for an additional minute until fragrant.
Add the chopped mushrooms to the skillet along with the dried thyme, salt, and pepper. Cook for about 6-8 minutes, stirring occasionally, until the mushrooms release their moisture and begin to brown.
In a mixing bowl, combine the softened cream cheese, shredded mozzarella cheese, grated Parmesan cheese, and sour cream. Mix until well combined.
Add the cooked mushroom mixture to the cheese mixture and stir until evenly distributed.
Transfer the dip to a baking dish and spread it evenly.
Bake in a preheated oven at 375°F (190°C) for about 15-20 minutes, or until bubbly and golden on top.
Remove from the oven, let cool slightly, and garnish with chopped fresh parsley before serving.