Mix matzo-ball ingredients and chill, covered, 30 minutes to 2 hours.
Preheat oven to 400°. Cut top off garlic, sprinkle with salt and pepper, and drizzle with some oil. Wrap in foil; roast 1 hour.
In a large pot, heat 1 tbsp. oil over medium heat. Add vegetables and 1/2 tsp. salt, cover, and cook until starting to soften, 10 minutes. Squeeze garlic from cloves into a bowl, whisk in 1/2 cup stock, and pour into vegetables with remaining stock.
About 30 minutes before serving, bring to a boil, then lower to a simmer. Using wet hands, loosely form matzo mixture into scant 1-in. balls; add to stock as you go. Simmer until cooked through, 30 minutes. Season with salt and pepper.
Serve hot, sprinkled with parsley.
Make ahead: Garlic, up to 1 week, chilled. Soup through step 3, up to 2 days, chilled.