Bring a large pot of salted water to a boil and cook broccoli florets until they start to soften, but are still bright green (about 4-5 minutes). Drain broccoli, but be sure to reserve about ¾ cup of the water.
Put broccoli in a blender along with ¾ cup of reserved water, vegetable or chicken broth, Greek yogurt, 2 teaspoon minced garlic, 2 teaspoon lemon juice, ½ teaspoon red pepper flakes, ½ teaspoon salt, and ¼ teaspoon pepper and puree until smooth.
Heat olive oil in a medium saucepan over medium heat and add mushrooms, remaining 1 teaspoon minced garlic, remaining 2 teaspoon lemon juice, and remaining ½ teaspoon red pepper flakes. Cook for about 4 minutes. Season with about ¼ teaspoon salt and ¼ teaspoon pepper. Add parsley and toss.
Serve broccoli soup in bowls and top with sautéed mushrooms. Sprinkle a little parmesan cheese on top, if desired.