In a large pot of boiling salted water, cook the cheese tortellini according to package instructions until al dente. Drain and set aside.
In a large skillet, heat the olive oil over medium heat. Add the chopped onion and cook until translucent, about 3-4 minutes.
Stir in the sliced mushrooms and minced garlic. Cook for an additional 5-7 minutes until the mushrooms are tender and browned.
Reduce the heat to low and pour the heavy cream into the skillet. Stir well, and then add in the grated Parmesan cheese and thyme. Allow the sauce to simmer for about 3-4 minutes, stirring occasionally, until it thickens slightly.
Add the cooked tortellini to the skillet. Toss gently to coat the pasta evenly with the creamy mushroom sauce. Season with salt and pepper to taste.
Plate the creamy mushroom tortellini, garnishing with fresh parsley on top for a pop of color.