Remove any tough outer leaves from the cabbages. Quarter the green cabbage and red cabbage, remove cores, then slice each quarter into very thin shreds (about 1/8 inch) using a sharp knife or mandoline. Transfer shredded cabbage to a large mixing bowl.

Shred carrots and onion

Peel the carrots and grate them on the large holes of a box grater or use a julienne peeler; add to the cabbage. Thinly slice the red onion into half-moons and add to the bowl.

Add the chopped fresh parsley, 1 tsp kosher salt and 1/2 tsp black pepper to the vegetables. Toss everything together thoroughly to distribute salt which will slightly soften the cabbage while you prepare the dressing.

Dressing

Combine dressing ingredients

In a medium bowl or jar, whisk together 1/2 cup mayonnaise, 1/4 cup Greek yogurt, 2 tbsp apple cider vinegar, 1 tbsp fresh lemon juice, 1 tsp honey, 1 tsp Dijon mustard, 1 tsp granulated sugar, 1/2 tsp celery seed, 1/4 tsp black pepper and 1/4 tsp kosher salt until smooth and homogenous.

Adjust seasoning

Taste the dressing and adjust acidity or sweetness as needed: add an extra 1/2 tsp honey for more sweetness or 1/2 tsp vinegar or lemon for more tang. The dressing should be creamy, slightly tangy, and slightly sweet.

Dress the slaw

Pour the dressing over the shredded vegetables. Using two large forks or tongs, toss thoroughly to coat all pieces evenly. Continue tossing 1–2 minutes until dressing is evenly distributed and cabbage begins to soften slightly.

Chill and rest

Cover the bowl and refrigerate the slaw for at least 30 minutes (preferably 1 hour) to let flavors meld and the cabbage soften. Toss once more just before serving and adjust salt and pepper to taste.