Melt butter in a stock pot over medium-low heat. Cook and stir onions, carrots, celery, and garlic in hot butter until the onions are translucent, 5 to 10 minutes. Add cauliflower and zucchini to onion mixture; cook until just warmed, 1 to 2 minutes.
Stir potatoes and broth into vegetable mixture; bring to a boil. Reduce heat and simmer until potatoes are just tender, 15 to 20 minutes. Add kale to soup; cook until kale is slightly wilted, 2 to 3 minutes.
Pour milk into a blender and ladle about 2/3 the soup into blender, working in batches if needed. Cover and hold lid down; pulse a few times before leaving on to blend until smooth, 1 to 2 minutes.
Pour pureed soup back into stock pot; stir in corn, paprika, dill, salt, and pepper. Cook soup until corn is warmed, 2 to 3 minutes. Serve soup garnished with Cheddar cheese and parsley.