Trim any damaged outer leaves from the cabbages. Cut each cabbage into quarters, remove the core, and thinly shred using a sharp knife, mandoline, or food processor fitted with the shredding disc. Place the shredded green and red cabbage in a large mixing bowl.

Prepare carrots and onion

Peel (if desired) and grate the carrot on the large holes of a box grater or using a food processor. Thinly slice the red onion into half-moons. Add the grated carrot and sliced onion to the bowl with the cabbage. Add the chopped parsley.

Lightly toss the vegetables together so they are evenly distributed and set aside while you make the dressing. Letting them rest 10–15 minutes will soften them and allow flavors to marry.

Dressing

Whisk dressing

In a medium bowl, combine the mayonnaise, plain yogurt or sour cream, apple cider vinegar, Dijon mustard, granulated sugar, celery seed, kosher salt, black pepper, and optional lemon juice. Whisk thoroughly until smooth and the sugar is dissolved. Taste and adjust seasoning — add more salt, pepper, or vinegar if you prefer more tang.

Combine slaw and dressing

Pour the dressing over the cabbage mixture. Using two large forks or salad tongs, toss gently but thoroughly until all vegetables are evenly coated. If the slaw seems dry, add up to 1–2 tablespoons of water or additional yogurt to reach desired consistency.

Chill and rest

Cover the bowl and refrigerate for at least 30 minutes, preferably 1 hour, to allow flavors to meld and the cabbage to soften slightly. Stir again just before serving and adjust seasoning to taste.

Serve chilled as a side for sandwiches, BBQ, or as a topping for tacos. Keeps well in an airtight container in the refrigerator for 3–4 days; the texture will soften over time.