Trim outer leaves from the cabbages. Cut each cabbage into quarters, remove the core, and slice very thinly to yield about 6 cups green and 2 cups red shredded cabbage. Peel and grate the carrots to make about 1½ cups. Place shredded cabbage and grated carrot into a large mixing bowl.

Add the sliced green onion and chopped parsley (if using) to the bowl with the vegetables and toss gently to combine.

Dressing

In a medium bowl, whisk together the mayonnaise, yogurt (or sour cream), apple cider vinegar, sugar, Dijon mustard, celery seed, lemon juice (if using), salt, and black pepper until smooth and sugar has mostly dissolved. Taste and adjust seasoning: add more sugar for sweetness, vinegar or lemon for tang, or salt/pepper to taste.

Assemble and Rest

Pour the dressing over the shredded cabbage mixture. Using two large spoons (or clean hands), toss thoroughly until the vegetables are evenly coated with dressing.

Cover the bowl and refrigerate at least 1 hour (ideally 2–4 hours) to allow flavors to meld and cabbage to soften slightly. If serving immediately, let rest 15–20 minutes at minimum.

Before serving, taste and adjust seasoning — add a pinch more salt, pepper, or a splash of vinegar or lemon juice if needed. Toss again and serve chilled.