Cook pasta according to package directions, omitting salt and fat. Reserve 1/2 cup cooking liquid; drain pasta. While pasta cooks, remove stems from portobello and shiitake mushrooms; reserve for another use (such as making stock). Scrape out black gills from underside of portobello caps with a spoon; discard gills. Cut each portobello cap in half; cut each half crosswise into 1/2-inch slices. Cut shiitake caps into 1/2-inch slices. Heat oil in a large skillet over medium-high. Add thyme and garlic; cook, stirring constantly, until fragrant, about 30 seconds. Add portobello and cremini mushrooms; cook, stirring often, 3 minutes. Add shiitakes; cook, stirring occasionally, until tender and liquid has evaporated, about 5 minutes. Stir in soy sauce; cook until absorbed. Stir in crème fraîche, salt, pepper, and 1/4 cup reserved cooking liquid. Add pasta; toss gently to coat. Stir in remaining cooking liquid as needed to reach desired consistency. Sprinkle with parsley.