4 cocktail umbrellas, for garnish

To make the ice cream, in a large, heavy saucepan, combine the cream, coconut milk, whole milk, rum, and sugar. Bring to a simmer over medium heat, stirring to dissolve the sugar. Remove from the heat. In a medium bowl, beat the eggs until pale yellow and frothy. Add about 1/2 cup of the hot cream mixture, and whisk to combine. Add the egg mixture to the saucepan with the remaining hot cream and whisk.

Return to medium heat and cook, stirring constantly with a heavy wooden spoon until thick enough to coat the back of a spoon, 4 to 5 minutes.

Remove from the heat and strain through a fine mesh strainer into a clean container. Refrigerate until well chilled, at least 3 hours. Pour into an ice cream maker and process according to the manufacturer's instructions. Transfer to a plastic container and freeze until ready to serve. Heat a large, heavy skillet over medium-high heat. Add the butter and when melted, add the pineapple. Cook, stirring, for 3 minutes. Sprinkle the sugar over the pineapples and continue to cook until the sugar has completely dissolved and starts to bubble, about 1 minute. Remove the pan from the heat and add the rum. Return to the heat and carefully tip the pan to ignite the alcohol (or, alternately, light with a match). Remove from the heat when the flames extinguish.

To assemble the dessert, spoon 2 scoops of coconut ice cream in each of 4 frosted parfait glasses. Top each portion with 1/2 cup of the pineapple and the glaze. Sprinkle 1 tablespoon of the toasted coconut into each glass, and garnish with an orange wedge, maraschino cherry, and a cocktail umbrella.