Combine potatoes, onion, celery, 1 tablespoon salt, and thyme in a saucepan. Cover with chicken broth. Bring to a boil over high heat. Reduce heat to medium-low and simmer until potatoes are tender, about 15 minutes. Add milk, butter, corn, and 1/4 teaspoon black pepper. Cook and stir gently until butter is melted, about 30 seconds.
Place flour, parsley, 1/2 teaspoon salt, and 1 dash pepper in a bowl. Crack egg into the bowl and combine with fingers or a fork to create very small dumplings or rivels. Add rivel mixture to the soup. Stir to combine.
Reduce heat to low and stir occasionally, until rivels are cooked and soup is thickened, about 10 minutes more.
