Arrange rack at center position and preheat oven to 450 degrees. Oil a large, heavy baking sheet generously with olive oil.
Cut each eggplant into 1-inch cubes and place them on baking sheet. Drizzle with 1/2 cup olive oil, then toss to coat all the eggplant well. Spread diced eggplant in a single layer in pan and season with 1 teaspoon salt. Roast, stirring every 5 minutes, until golden and tender, about 15 minutes. Watch carefully so that eggplant does not overcook. Remove and cool 10 minutes. Transfer to large non-reactive mixing bowl and add tomatoes, onion and garlic.
Whisk lemon juice with cumin, paprika and cayenne pepper in a small bowl. Pour mixture over eggplant and tomatoes, and toss very gently, taking care to keep eggplant intact.
Add parsley to bowl with eggplant. Mix gently. Taste and season with more salt, and a squeeze or two more of lemon juice if desired. (This salad actually tastes better when prepared 3 to 4 hours ahead so that the flavors can meld. It can be prepared a day in advance. Refrigerate and bring to room temperature 30 minutes before serving when making ahead.).
Spoon salad into a serving bowl or onto a platter. Makes 4 to 6 servings.