Wash the broccoli under cold water, chop into florets, dice the onion, and mince the garlic.

Heat olive oil in a large pot over medium heat. Sauté onion for about 5 minutes until translucent, then add garlic for 1 minute.

Add diced potatoes and carrot to the pot, cooking for about 5 minutes until they begin to soften.

Pour in vegetable broth, bring to a gentle boil, then add broccoli, salt, and pepper. Simmer for 15-20 minutes until tender.

Blend the soup until smooth using an immersion blender or transfer to a countertop blender in batches.

Return to pot, adjust consistency with more broth or water if needed, and check seasoning.

Ladle soup into bowls, garnish with Parmesan, red pepper flakes, and parsley, then serve hot.