Bring the milk to a simmer in a heavy-bottomed saucepan. Stirring constantly, pour in the cornmeal in a thin stream until you have a smooth mush.
Off heat, beat in the butter, salt, and corn, then the eggs which have been beaten very light. Add the parsley. If you wish to take the trouble,the spoon bread is even better when the egg yolks are added first, then the whites beaten fairly stiff and folded in. More trouble, though.
Pour batter into a shallow, greased two-quart casserole. Bake in the center of the oven about 45 minutes. If the top browns too fast, place the dish lower in the oven.