Trim and remove any tough outer leaves from the cabbages. Cut cabbages into quarters, remove cores, and finely shred using a sharp knife, mandoline set to a thin setting, or food processor with shredding disk. Place shredded green and red cabbage in a large mixing bowl.

Add carrots and onion

Peel and grate or julienne the carrots, then add to the bowl with the cabbage. Thinly slice the red onion into half-moons and add. Toss in the chopped parsley.

Season and rest (optional)

Sprinkle the kosher salt and half of the black pepper over the shredded vegetables. Toss thoroughly to distribute. Let the vegetables sit for 10 minutes (this helps draw out a little moisture and meld flavors). If large amounts of liquid accumulate, gently drain or blot with a paper towel.

Dressing

Make the dressing

In a medium bowl or jar, combine mayonnaise, plain yogurt or sour cream, apple cider vinegar, sugar or honey, Dijon mustard, celery seed, lemon juice, and white pepper if using. Whisk or shake vigorously until smooth and slightly glossy, taste and adjust sweetness, acidity, or salt as needed.

Combine dressing with salad

Pour about three-quarters of the dressing over the shredded vegetables and toss thoroughly with tongs or two large spoons until evenly coated. Add more dressing a little at a time until you reach the desired creaminess (reserve any extra dressing for serving).

Adjust seasoning and chill

Taste and adjust seasoning with remaining black pepper and a pinch more salt, sugar, or vinegar to balance. For best flavor, cover and refrigerate the coleslaw for at least 30 minutes to 1 hour to let flavors meld. Give a final toss before serving and add a small squeeze of fresh lemon if desired.

Storage

Store leftover coleslaw in an airtight container in the refrigerator for up to 3 days. Note that the salad will soften as it sits; you can refresh with a little extra lemon juice or a tablespoon of mayonnaise before serving.