Place the yogurt in a clean tea towel, gather up the edges, suspend over a bowl, and allow liquid to drain for about 2 hours in the refrigerator.
Use another clean tea towel and place the chopped cucumber in it. Squeeze all the liquid out that you can and discard.
Bring quinoa and water to a boil in a small saucepan. Cover, reduce heat, and simmer until liquid is absorbed. Stir occasionally to be sure quinoa does not stick to the bottom and burn. Let cool 10 minutes.
In a food processor, pulse onions and parsley so that they are finely chopped. Add in the remainder of the meatball ingredients, except the oil. Add in the cooked quinoa and gently pulse just to combine. Do not overprocess.
Preheat oven to 350 degrees F. Form about 42 meatballs out of a level tablespoon. Heat oil in a skillet, over medium heat, and brown meatballs on all sides in batches. If the oil starts to smoke, reduce heat a bit.
Transfer meatballs to a sheet pan and roast to an internal temperature of 160 degrees F. This will take about 10 to 15 minutes.
In a small saucepan, boil pomegranate juice until it reduces to about 3 tablespoons of syrup, about 10 minutes. Cover and keep warm.
In a bowl, combine the drained yogurt, squeezed cucumber, and salt. Drizzle with the pomegranate reduction and lightly swirl to make pretty. Sprinkle with pomegranate seeds, if using, or add mint leaves. Serve with meatballs.