Rinse cabbages, carrot, and parsley under cold water. Pat dry. Remove any wilted outer leaves from cabbages. Cut each cabbage into quarters, remove cores, then thinly slice into fine shreds (about 4 cups green and 2 cups red). Peel and shred the carrot to yield about 1 cup. Thinly slice half a cup of red onion and roughly chop 2 tablespoons of parsley.

Place the shredded green cabbage, red cabbage, shredded carrot, sliced red onion, and chopped parsley in a large mixing bowl. Toss lightly to combine. If you prefer a slightly softer slaw, sprinkle 1/2 teaspoon salt over the vegetables, toss, and let sit 10 minutes, then drain any released liquid and proceed (optional).

Make dressing

Whisk dressing

In a medium bowl, combine 1/2 cup mayonnaise, 1/4 cup plain Greek yogurt, 2 tablespoons apple cider vinegar, 1 teaspoon Dijon mustard, 1 tablespoon granulated sugar, 1/2 teaspoon celery seed, 1 teaspoon kosher salt, and 1/2 teaspoon freshly ground black pepper. Whisk until smooth and sugar is dissolved. If the dressing is too thick for your preference, whisk in up to 2 tablespoons water, 1 tablespoon at a time, until it reaches a creamy pourable consistency.

Taste the dressing and adjust seasoning: add a little more vinegar if you want brighter acidity, more sugar for sweetness, or more salt and pepper to balance.

Combine and finish

Pour the dressing over the prepared vegetables. Using two large spoons or salad tongs, toss thoroughly until all vegetables are evenly coated. Use gentle pressing and folding motions to distribute the dressing without bruising the cabbage.

Cover the bowl and refrigerate for at least 30 minutes to allow flavors to meld. For best flavor and texture, refrigerate 2–4 hours. Before serving, taste and adjust seasoning (add salt, pepper or a splash of vinegar if needed).

Serve chilled as a side dish. Keeps well covered in the refrigerator for up to 3 days; toss once before serving as some liquid may collect.