Using a grapefruit spoon or small knife, remove pulp and seeds from tomatoes. Let tomato shells drain, cut side down, on paper towels.
Drop garlic cloves, anchovy fillets, and capers into a food processor with the motor running and whirl until finely chopped. Add 1 1/4 cups parsley leaves and the goat cheese; whirl until parsley is finely chopped. Season with salt to taste, and whirl in vinegar if it seems to need more zing.
Mound tomato halves with goat cheese filling and top some or all with small parsley leaves.
MAKE AHEAD: Chilled airtight, up to 2 hours.