Step 1: Set up a charcoal grill for direct grilling and preheat to high.
Step 2: When ready to cook, prick the eggplants in a few spots with a fork. Place them directly in the embers and grill until the skins are charred all over and the flesh is very soft, 5 to 8 minutes per side (20 to 32 minutes in all). Transfer the grilled eggplants to a plate to cool.
Step 3: Scrape any really burnt skin off the eggplants, but leave some of it on; the dark spots will add color and character. Coarsely puree the eggplants and garlic in a food processor (or chop them by hand or mash them with a fork). Add the tahini, if using, 2 tablespoons of the olive oil, the lemon juice, salt, and pepper and pulse just to mix. Add the parsley and process in short bursts, just to mix (overprocessing will turn the dip green). Taste for seasoning, adding more salt, pepper, and/or lemon juice as necessary; the dip should be very flavorful. Transfer the dip to a shallow nonreactive bowl, drizzle with the remaining 1 tablespoon of oil, and sprinkle with paprika. Serve with lemon wedges and grilled pita bread.