In a mixing bowl, whisk together eggs, egg white and salt. Whisk in matzo meal, oil and 3 Tbsp cold water.
Cover and chill the mixture at least 1 hour or overnight.
In a large pot, bring chicken broth to a boil. Add parsnips, carrots and onions, reduce the heat to medium-low and cook for 5 minutes.
Gently roll the chilled matzo dough by level teaspoonfuls into balls, dropping them into the simmering broth as you work. Cook the matzo balls, covered, for 15 minutes. Do not lift the lid; the broth must simmer rapidly to allow the matzo balls to expand properly.
Uncover the pot, add broccoli and mushrooms and simmer until the broccoli is just tender, 2 to 3 minutes.
Ladle into bowls, sprinkle with dill or parsley and serve.