2/3 cup sweetened cream of coconut, such as Coco Lopez, at room temperature
1 ripe medium pineapple (about 3 1/2 pounds)
1/2 cup light brown sugar
1 tablespoon unsalted butter
Set a 2-quart container and 6 parfait glasses in the freezer. In a medium bowl, stir together the pineapple juice and cream of coconut. Working in 2 batches, freeze the mixture in an ice cream machine according to the manufacturer's instructions. Transfer the sorbet to the chilled container, cover tightly and freeze.
Using a long sharp knife, remove the skin from the pineapple. Cut the pineapple in half lengthwise and remove the core. Cut one pineapple half into 1/4-inch dice. Cut three 1/3-inch thick crosswise slices from the other pineapple half, then cut 2 neat triangles from each slice. Reserve the remaining pineapple for another use.
In a large, heavy, nonreactive skillet, combine the brown sugar and butter with 1 tablespoon of water and stir over moderately high heat until bubbling. Add the 6 pineapple triangles and cook until golden brown, 3 to 4 minutes per side. Transfer the triangles to a plate to cool.
Add the diced pineapple to the skillet and cook until the juices evaporate, about 5 minutes. Transfer to a bowl and let cool.
Place a scoop of slightly softened sorbet in each of the 6 chilled parfait glasses. Cover the sorbet with half of the diced pineapple. Make a second layer with another scoop of sorbet and the remaining diced pineapple. Top the parfaits with the remaining sorbet. Just before serving, garnish each parfait with a caramelized pineapple triangle and a Coconut Tuiles. Pass the remaining Coconut Tuiles separately.
The pineapple can be prepared up to 4 hours ahead. The parfaits can be assembled up to 30 minutes ahead and held in the freezer.