Pierce the eggplant in a few places, then grill the whole eggplant for about 30 to 40 minutes over medium hot coals, turning about every 7 minutes. (Instead of grilling, you can roast the eggplant in a 400° F/220° C oven.) Peel the skin off while it's still hot and remove the stem end, then rough chop the flesh. Place it in a colander and drain out the liquid.
Add the flesh to the work bowl of the food processor and puree it. Add the lemon juice and tahini and pulse a few times until well mixed.
Add the garlic and 1 teaspoon of salt. Process well and add more lemon juice and salt to taste.
Add the olive oil and parsley, processing until the parsley is chopped but still visible.
Serve in a shallow dish garnished with parsley. Will keep in the refrigerator for up to 3 days. Bring to room temperature before serving.