Heat the olive oil in a large pot over medium heat. Add the onion and garlic and cook until softened, about 5 minutes.
Add the cauliflower florets, vegetable broth, milk, dried thyme, and smoked paprika to the pot. Bring to a boil, then reduce heat and simmer for about 20 minutes, until the cauliflower is tender.
Use an immersion blender to blend the soup until smooth and creamy. Alternatively, transfer the soup to a blender and blend in batches.
Stir in the shredded cheddar cheese until melted and creamy. Season with salt and pepper to taste.
Serve the chowder hot, garnished with crumbled bacon and fresh parsley.