Peel and seed melons, then cut flesh into 1-inch cubes. Arrange, separated, in a single layer on a shallow baking sheet lined with wax paper. Freeze until firm, at least 1 hour. (If freezing for more than 1 hour, cover with plastic wrap first.)
Reserve some strawberries for garnish. Slice remaining berries, toss in a bowl with sugar, and let stand.
Grind frozen cantaloupe in a food processor until it resembles fine crumbs; transfer to a stainless steel bowl. Grind frozen honeydew into crumbs; pour into another bowl.
Purée sugared strawberries in a food processor. Transfer to a small serving bowl.
Slice reserved strawberries. Scoop frozen melon with an ice cream scoop and put in parfait glasses. Garnish with sliced berries, and serve with strawberry sauce.