Cook the tortellini according to package instructions until al dente. Drain, rinse with cold water to stop cooking, and set aside.
While the tortellini cooks, heat a large skillet over medium-high heat and add the butter.
Add the sliced mushrooms and cook on high/medium-high for about 8–10 minutes, stirring occasionally, until their liquid evaporates and they start to brown.
Reduce heat to low, pour in the heavy cream, stir in the dried thyme and salt, and bring the mixture to a gentle boil.
Add the drained tortellini to the skillet and toss to combine evenly with the mushroom-cream sauce.
Sprinkle the shredded Mexican cheese blend evenly over the tortellini, cover with a tight-fitting lid, and cook for 5 minutes, until the cheese is melted.
Remove the lid, garnish with chopped parsley, and serve immediately.