Butter a 2 1/2 quart [2 1/2 lt] baking pan with a tablespoon of butter (see notes).
Heat oven to 350 ºF [175 ºC].
Mix coconut milk and broth, salt to taste. Mix in parsley, cornmeal and the remaining butter. Stir the batter with a spatula and pour into a 3 qrt [3 lt] cooking pot and heat on the stove over medium heat, stirring constantly to avoid sticking.
When it breaks the boil, lower the heat and continue stirring until it thickens enough to start lifting from the pot when you stir and it sticks to the spoon.
Pour batter into the pan and bake until you insert a knife in the center and it comes out clean (30-40 minutes). It should be light golden brown. Let it cool down to room temperature before removing from the pan.
Serve alone as a snack, with pollo guisado, or your choice of stewed meat.
To obtain the traditional pot-shaped cake, bake it in a enamel cast-iron pot (dutch oven). The disadvantage is that you might have a hard time getting it out in one piece. If you don't want to take that risk, bake in a nonstick baking pan, the deepest you have.Cornmeal is slightly finer than polenta. If you find it impossible to find cornmeal, use polenta, but be aware that the texture will be a bit "grainier" than with regular cornmeal.