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 tsp minced garlic, 2 tsp lemon juice, ½ tsp red pepper flakes, ½ tsp salt, and ¼ tsp pepper and puree until smooth.
Heat olive oil in a medium saucepan over medium heat and add mushrooms, remaining 1 tsp minced garlic, remaining 2 tsp lemon juice, and remaining ½ tsp red pepper flakes. Cook for about 4 minutes. Season with about ¼ tsp salt and ¼ tsp pepper. Add parsley and toss.
Serve broccoli soup in bowls and top with sautéed mushrooms. Sprinkle a little parmesan cheese on top, if desired.