To make the dumplings: In a bowl, stir together flour and 1/2 teaspoon salt. Add 2 tablespoons butter and, using your fingers, rub into flour until pea-size crumbles form. Add egg and stir until dough forms. Refrigerate until ready to use.
Heat remaining 4 tablespoons butter in a 6-quart saucepan over medium-high heat. Add paprika and onion and cook, stirring, until soft, about 5 minutes. Add stock, cauliflower and carrot, season with salt and pepper, and bring to a boil. Reduce heat to medium and cook, stirring occasionally, until vegetables are tender, about 15 minutes.
Using a 1/2 teaspoon measuring spoon, portion out and drop all dumpling dough into simmering soup. Cook, stirring occasionally, until dumplings are cooked through, about 3 minutes. Season with salt and pepper.
To serve, ladle soup and dumplings into serving bowls and garnish with parsley.