Stir onion, vinegar, garlic, and brown sugar together in 6 quart slow-cooker. Spread whole grain mustard over all chicken pieces. Place chicken, breast-side down, in slow-cooker. Cover and cook on low for 5-6 hours, until chicken is fork tender.
Remove chicken from slow-cooker and let cool for 5 minutes.
Pour braising liquid through sieve. Transfer onion solids and 1/2 cup of the liquid to a large bowl. Add dressing ingredients (vinegar, yellow mustard, parsley, salt and pepper) to the onion mixture and whisk together.
Shred chicken in large pieces and toss with dressing.