Trim outer leaves from the cabbages. Quarter each cabbage, remove the core, and slice the quarters very thinly using a sharp knife or mandoline to yield about 4 cups green cabbage and 2 cups red cabbage. Transfer shredded cabbage to a large mixing bowl.

Add carrot and onion

Peel and grate (or julienne) enough carrots to make 1 cup. Add grated carrot to the bowl with cabbage. Thinly slice 1/4 cup red onion (about small half onion) and add it to the bowl. Add the chopped parsley and toss the vegetables lightly to combine.

Optional step — reduce crunch: if you prefer a softer coleslaw, sprinkle the shredded cabbage and carrot with 1 tablespoon of apple cider vinegar (from dressing amount) and 1/2 teaspoon salt, toss, and let sit 10 minutes, then drain any excess liquid before adding dressing.

Dressing

Make dressing

In a medium bowl, whisk together 1/2 cup mayonnaise, 1/4 cup plain Greek yogurt, 2 tablespoons apple cider vinegar, 1 teaspoon Dijon mustard, and 1 tablespoon honey until smooth. Whisk in 1/2 teaspoon celery seed (if using), 1 teaspoon salt, and 1/2 teaspoon freshly ground black pepper. Taste and adjust seasoning: add more vinegar for tang, honey for sweetness, or salt and pepper as needed.

Combine dressing and vegetables

Pour the dressing over the shredded cabbage mixture. Using tongs or two large spoons, toss thoroughly until all vegetables are evenly coated. Mix gently to avoid bruising the cabbage.

Chill and marry flavors

Cover the bowl and refrigerate the coleslaw for at least 30 minutes (preferably 1 hour) to let flavors meld. Stir again just before serving and adjust seasoning with a pinch more salt or pepper if needed.

Serve: transfer to a serving bowl or platter. Coleslaw keeps well refrigerated in an airtight container for up to 3 days; if it sits longer, you may need to refresh seasoning and texture with a squeeze of vinegar and a stir.