Preheat oven to 320 degrees. Prepare a round cake pan with butter.
In a saute pan over medium heat, add apples, butter and sugar. Stir with a wooden spoon for about 5 to 6 minutes, until it turns a golden caramel color. Remove from heat and stir in the rum. Pour into prepared cake pan and smooth the top.
In a mixing bowl, add the eggs, egg yolk and sugar. Whisk until combined. Gradually stir in the milk. Pour into the cake pan through a sieve. Bake for 40 minutes.
Cool in the refrigerator. Turn upside down onto a plate and decorate with whipped cream and parsley.