Preheat the broiler. Trim the stem ends of the eggplant and make 4 small slits in each. Broil the eggplant 2 or 3 inches from heat source, turning, until evenly browned and soft in the center, about 15 minutes. Cool.
Using a teaspoon, scrape the flesh away from the skins and remove as many seeds as possible. Squeeze the flesh between paper towels to remove excess moisture.
Place the onions, garlic, and vinegar in a food processor and process until finely chopped. Add the eggplant, yogurt or mayonnaise, and parsley, and pulse three times. Add the oil in a stream while pulsing until almost smooth (do not overmix). Taste and season with salt and pepper to taste, and add olive oil and vinegar if desired.
Makes about 3 cups.