Shuck oysters and set aside, reserving 12 shells (large “cup” sides). In a small bowl, combine mascarpone, chives, and parsley. Season with salt and pepper to taste. (Go easy on the salt: The oysters and prosciutto are salty, too.)
Position oven rack 4 to 5 inches from broiler and turn it on. Cut prosciutto slices in thirds (for a total of 12 4-by-2-inch pieces) and lay them on a clean surface. Place one oyster on bottom third of prosciutto piece; then place a small dollop (1 to 2 teaspoons) of mascarpone on top of oyster. Fold prosciutto over and place seam side down into reserved shell. Repeat with remaining oysters.
Arrange shells on a salt-lined baking sheet. Broil about 1 minute or until prosciutto crisps and lightly browns. Garnish with chopped parsley and serve immediately.