Trim core from both cabbages. Shred green and red cabbage finely (about 6 cups green, 2 cups red). Peel and grate carrots to measure 1 1/2 cups. Thinly slice red onion into very thin half-moons (about 1/4 cup). Place cabbage, carrot, and onion in a large mixing bowl. Add chopped parsley, celery seed, 1 teaspoon salt, and 1/2 teaspoon black pepper. Toss gently to combine.

Make Dressing

Whisk dressing

In a medium bowl, combine 3 tablespoons mayonnaise, 3 tablespoons Greek yogurt (or sour cream), 2 tablespoons apple cider vinegar, 1 tablespoon lemon juice, 1 tablespoon granulated sugar, 1 teaspoon Dijon mustard, 1 teaspoon honey, 1/4 teaspoon black pepper, and 1/4 teaspoon salt. Whisk until smooth and sugar is dissolved. Taste and adjust acidity or sweetness as needed (add up to 1 teaspoon more sugar or 1/2 teaspoon more vinegar).

Combine and Chill

Dress the slaw

Pour the dressing over the shredded vegetable mixture. Using two large forks or salad tongs, toss thoroughly until all ingredients are evenly coated. Let the slaw sit at room temperature for 10 minutes, then transfer to the refrigerator and chill for at least 30 minutes to allow flavors to meld. For best texture and flavor, chill 1–2 hours.

Final seasoning

After chilling, taste and adjust seasoning with additional salt, pepper, or a splash of vinegar if needed. If the slaw has released too much liquid, gently drain a little before serving to avoid a watery dish. Serve cold as a side or topping.