Preheat the oven to 325 degrees F (160 degrees C). Rub the insides of four 5 1/2-ounce ramekins with oil.
Melt 1/2 cup plus 2 tablespoons sugar, without stirring, in a skillet over medium heat. Swirl the skillet occasionally until sugar caramelizes to an amber color. Quickly pour caramel into each ramekin to cover the bottoms. Place ramekins in a baking dish.
Combine cream cheese, Parmesan cheese, vanilla extract, salt, and eggs in a bowl. Whisk until smooth.
Heat 1 cup plus 2 tablespoons milk until almost simmering. Stir in sugar and pour into the cream cheese mixture.
Ladle mixture into the ramekins. Fill the baking dish with enough hot water to come halfway up the sides of the ramekins.
Bake in the preheated oven until custard is just barely set, 45 to 60 minutes. When shaken, custard should jiggle as one mass. Remove ramekins from baking dish and let cool to room temperature.
Mix graham cracker crumbs with melted butter and spread some of the mixture over each flan. Refrigerate until fully chilled, at least 2 hours.
Run a knife along the edge of each chilled flan and invert onto a plate. Drizzle any remaining caramel on top.