Preheat the oven to 350 degrees F. Line a baking sheet with aluminum foil.
Wipe the mushrooms with a damp paper towel and remove the stems. Finely mince the stems. Set the caps aside.
In a large skillet, heat 1 tablespoon of the oil over medium-high heat. Add the chopped mushroom stems, onion, parsley, and thyme and cook until the vegetables are translucent, about 2 minutes. Add the sausage and cook, using a wooden spoon to stir and break up any clumps, until the sausage has browned, about 4 minutes. Reduce the heat to low and stir in the cream cheese and half of the Parmesan, stirring until the cheese melts—some remaining lumps are fine. Set aside.
Place the mushroom caps in a medium bowl and toss with the remaining 1 tablespoon oil to coat. Arrange the caps, open side up, on the prepared baking sheet. Use a teaspoon to spoon the filling into the caps, shaping the tops into mounds. Sprinkle with the remaining Parmesan and bake for about 20 minutes, until the tops are lightly browned and the filling is heated through. Arrange on a platter, garnish with parsley, and serve warm.
