Prepare the vegetables: Finely shred the green cabbage, red cabbage, and carrot. Add them to a large mixing bowl with fresh parsley.
Make the dressing: In a separate bowl, whisk together mayonnaise, sour cream, Dijon mustard, apple cider vinegar, lemon juice, sugar, salt, and black pepper until smooth and creamy.
Combine: Pour the dressing over the shredded vegetables.
Toss: Gently toss everything together with salad tongs until the cabbage and carrots are evenly coated.
Chill: Cover and refrigerate for at least 30 minutes to allow the flavors to blend.
Serve: Give the coleslaw one final toss, garnish with fresh parsley, and serve chilled.