Make the dressing by whisking together ⅓ cup vegan mayo, ⅓ cup Greek yogurt, 2 tablespoons apple cider vinegar, 1 tablespoon mustard, 1 tablespoon sugar, 1 teaspoon salt, and ¼ teaspoon black pepper. Set aside.
Cut the red cabbage in half, then in half again to have four pieces.Thinly slice the cabbage with a sharp knife, a mandoline slicer, or a food processor with a slicing disk attachment. You need about 5 cups (15 ounces) red cabbage.
To a large mixing bowl, add red cabbage, 2 carrots (grated), 1 handful parsley (chopped)
Toss until well combined and serve immediately or store in the fridge for up to 3 days.