Preheat oven to 250°. Brush mushrooms with 1 tsp. olive oil and put, top side down, on a baking sheet. Bake until shrunken slightly, about 30 minutes.
In a medium bowl, toss parsley leaves with 1 tsp. oil and 1/2 tsp. vinegar. Add coarse salt to taste.
To serve, divide salad among four plates. Slice portabellas and arrange on salad. Drizzle with remaining olive oil and vinegar. Sprinkle with salt and add parmesan curls on the side.
Note: Nutritional analysis is per serving; Cholesterol data not available.