Start by melting butter in a large pot over medium heat. Add chopped onion and celery, cooking until they soften and become fragrant. Stir in the minced garlic and cook briefly to release its aroma without burning.
Sprinkle the flour into the pot and stir continuously for about a minute. This step forms a roux, which helps thicken the stew and gives it a smooth texture. Make sure there are no lumps before moving forward.
Slowly pour in the milk and broth while stirring constantly. This prevents clumps and ensures a creamy consistency. Let the mixture simmer gently until it begins to thicken slightly.
Stir in the corn, Old Bay seasoning, paprika, salt, and pepper. Allow the stew to cook for a few minutes so the flavors blend together. Taste and adjust seasoning if needed.
Gently fold in the crab meat, being careful not to break it apart too much. Let it heat through for about 5 minutes. Overcooking can make crab tough, so keep this step brief.
Turn off the heat and stir in fresh parsley for a burst of color and freshness. Let the stew sit for a couple of minutes before serving to allow flavors to settle.