Heat olive oil in a large saucepan over medium-high heat. Cook and stir shallot in hot oil for 30 seconds. Add mushrooms and continue cooking until they begin to brown. Stir in rice and 2 tablespoons butter; reduce heat to medium. Cook and stir until shallot and mushrooms turn golden brown.
Pour in wine; simmer until evaporated, stirring constantly. Season with thyme, celery salt, salt, and pepper. Reduce heat to medium-low; pour in 1/3 of the hot vegetable stock. Cook until evaporated, stirring constantly, about 7 minutes.
Pour in 1/2 of the remaining hot vegetable stock; continue to cook and stir until evaporated. Repeat once more with remaining hot stock. Stir constantly until rice is tender and creamy. Add hot water if needed to continue cooking until rice has softened.
Stir in Parmesan cheese, parsley, and remaining 2 tablespoons butter to serve.