Add whole zucchini to the grill and close the cover. Turn occasionally until zucchini are blackened and soft. Remove from heat and cool.
Peel each zucchini, placing flesh in a colander to drain. Drain for approx 1 hour. Don't discard the zucchini liquid quite yet - reserve some if needed for some extra flavor or liquid later on.
In a food processor or high-speed blender add drained zucchini, tahini, lemon juice, garlic, salt and olive oil. Blend until creamy. Taste for salt and consistency and add drained zucchini water as needed.
Spoon babagazuk onto a serving bowl and garnish with a drizzle of olive oil, sumac (or za'atar/aleppo pepper/urfa) and a few sprigs of parsley.
Serve room temp with crackers, toasted bread, pita or fresh veggies.