Hollow out each eggplant with a melon-ball cutter, working from bottom end and leaving about 1/3 inch eggplant flesh along interior walls.
Rinse rice in a sieve under cold water until water runs clear. Drain well.
Heat oil in a 12-inch heavy skillet over medium-high heat until it shimmers. Fry pine nuts, stirring frequently, until golden, about 3 minutes, then transfer with a slotted spoon to a bowl. Sauté onion and garlic, stirring occasionally, until golden, 6 to 8 minutes. Transfer 1/2 cup onion mixture to bowl with pine nuts. Add stock, tomatoes, 3/4 teaspoon salt, and 1/4 teaspoon pepper to skillet and simmer, uncovered, while stuffing eggplant.
Add rice, meat, allspice, 1 teaspoon salt, and 1/2 teaspoon pepper to bowl with onion mixture and mix well with your hands.
Stuff eggplant with meat mixture, being careful not to pack tightly (rice will expand during cooking). Transfer stuffed eggplants to skillet with tomato sauce and simmer, covered, carefully turning once, until rice is cooked through, 50 minutes to 1 hour (cut 1 in half to test).
If sauce is watery, transfer eggplant to a plate and boil sauce, stirring occasionally, until thickened slightly, 3 to 5 minutes, then adjust seasoning if necessary. Return stuffed eggplant to sauce. Squeeze lemon over dish and sprinkle with parsley before serving.