Heat the oven to 400°F (200°C). Halve each squash through the stem end and scoop out the seeds and strings. Trim a thin slice from the rounded bottom of each half so it sits flat.
Brush the cut sides and cavities with 1 tablespoon of the oil and season with salt and pepper. Place cut side down on a parchment-lined baking sheet and roast 25 minutes.
Turn the squash cut side up and roast 10–15 minutes more, until the flesh is very tender when pierced and the edges are browned.
Meanwhile, heat the remaining oil and the butter in a large skillet over medium-high heat. Add the mushrooms and cook, stirring only occasionally, until golden and the liquid has evaporated, 7–9 minutes.
Lower the heat to medium, add the shallot, garlic and thyme and cook 2 minutes. Pour in the cream and simmer until slightly thickened, about 2 minutes. Season with salt, pepper and lemon juice.
Spoon the mushrooms into the squash halves, sprinkle with Parmesan and return to the oven for 5 minutes. Garnish with parsley and serve.