Heat your barbecue grill to medium-high heat. Add wood chips to your grill or place the chips into a sealed foil pouch with slits for the smoke to escape.
Add a few slits into the eggplant especially in the wider end so that it will cook more evenly.
Place the whole eggplants on the grill (and right next to the foil pouch if using) and cook for about 30 minutes total, turning about every 10 minutes so the that the eggplant is wrinkled and soft on all sides.
Once the eggplants are cooked, set them into a bowl and cover with foil and let cool. Keep any juices collected in the bottom of the bowl. This is the reserved liquid that you will use to add to the dip.
Once cool enough to handle, remove the meat of the eggplant from the skin and place it into a strainer and let it drain. After about 10 minutes, add the drained eggplant into a food processor. Discard the drained liquid and the skins.
Add to the processor the lemon, garlic, tahini, reserved liquid, paprika, cumin, salt and pepper and pulse to combine the ingredients.
Scrape the dip into a serving bowl and drizzle with olive oil. Sprinkle with parsley and serve.