In a medium saucepan over medium heat, bring the stock to a simmer. Reduce the heat to low to keep warm.

In a large pot or Dutch oven, heat oil. Add onion and cook, stirring often, until softened, about 7 minutes.

Add mushrooms, garlic, thyme, bay leaf, and 1 Tbsp. butter. Cook, stirring frequently, until mushrooms are softened and golden brown, 10 to 15 more minutes; season with salt and pepper. Transfer the mushroom mixture to a medium bowl.

In same pot over medium heat, melt remaining 1 tbsp butter. Add rice and cook, stirring, until grains are coated and smell slightly toasty, about 2 minutes. Add wine and cook, stirring frequently, until wine has mostly absorbed, about 3 minutes.

Using a ladle, add about 1 cup of the stock. Cook, stirring often, until rice has mostly absorbed the liquid, 3 to 5 minutes. Continue to add the stock about 1 cup at a time, letting the rice absorb each addition of stock before adding more. Stir often and cook until risotto is al dente and creamy, not mushy, 20 to 25 minutes total. (You might not need all the broth.)

Return mushroom mixture to pot and stir to combine. Stir in Parmesan and peas until cheese melts. Top with parsley.