Remove outer leaves from cabbages. Cut cabbages in half, core, and finely shred using a sharp knife, mandoline, or food processor; measure 6 cups green and 2 cups red. Place shredded cabbage in a large mixing bowl.

Add carrots and onion

Peel and grate the carrot(s) to yield about 1 cup. Thinly slice 1 small red onion to yield 1/4 cup. Add grated carrot and sliced onion to the bowl with cabbage. Add chopped parsley and gently toss to combine.

Optional step to reduce wateriness: sprinkle 1/2 teaspoon kosher salt over the shredded vegetables, toss, and let sit 10 minutes, then gently squeeze out excess moisture and transfer to a clean bowl. (If you prefer a juicier slaw, skip this.)

Dressing

Mix dressing

In a medium bowl, whisk together 1/2 cup mayonnaise, 1/4 cup Greek yogurt (or sour cream), 2 tablespoons apple cider vinegar, 1 teaspoon Dijon mustard, 2 teaspoons granulated sugar, 1/2 teaspoon celery seed, 1 teaspoon kosher salt, 1/2 teaspoon freshly ground black pepper, and 1 teaspoon lemon juice (if using) until smooth and emulsified. Taste and adjust seasoning: add more vinegar for tang, sugar for sweetness, or salt/pepper as needed.

Combine and chill

Pour the dressing over the prepared slaw and toss thoroughly until all vegetables are evenly coated. Cover and refrigerate at least 30 minutes to let flavors meld and cabbage soften slightly. For best flavor, chill 2–4 hours. Before serving, taste and adjust seasoning; if mixture seems too thick, stir in 1–2 teaspoons water or additional vinegar to reach desired consistency.

Serve cold as a side for sandwiches, barbecues, or as a topping for tacos. Keeps in an airtight container in the refrigerator for 3–4 days; toss before serving as liquids may settle.