Prepare Soup: Tie half of dill and half of parsley in a bunch with kitchen string. Chop remaining dill and parsley to equal 2 tsp. each.
Bring chicken, next 4 ingredients, dill-parsley bunch, and 3 1/2 qt. water to a boil in a large Dutch oven over medium-high heat; skim any foam with a slotted spoon. Cover, reduce heat to medium-low, and simmer, 2 1/2 to 3 hours or until chicken is tender and falls off the bone.
Whisk together oil, eggs, and 1/4 cup water.
Add matzo meal and 1 3/4 tsp. kosher salt; whisk until well blended. Cover and chill 30 minutes.
Remove soup from heat. Skim fat from surface of broth.
Remove chicken and celery.
Pour broth through a fine wire-mesh strainer into a large bowl. Return broth, carrots, and parsnips to Dutch oven, discarding onion and herb bunch.
Let chicken, celery, and broth mixture cool 30 minutes.
Meanwhile, assemble and cook Matzo Balls: Shape matzo batter into 18 (1-inch) balls (about 1 Tbsp. each), using wet hands. Bring 2 1/2 qt. water to a boil in a large saucepan over medium-high heat. Drop matzo balls into boiling water; return to a boil. Cover, reduce heat to medium-low; simmer 30 minutes.
Remove matzo balls from water with a slotted spoon.
Squeeze juice from cooled celery ribs into broth. Discard celery ribs. Skin and bone chicken; shred chicken.
Add matzo balls, shredded chicken, lemon juice, kosher salt, and pepper to broth. Bring to a boil over medium-high heat. Reduce heat to medium-low; simmer 8 minutes. Stir in reserved dill and parsley; cook 2 minutes.
