Coat the slow cooker with melted butter. Add onion, celery, carrots, and garlic.
Season chicken with thyme, poultry seasoning, salt, and pepper. Nestle on top of the vegetables.
Whisk broth and condensed soup until smooth. Pour over the chicken and vegetables.
Cover and cook on High 4 hours or Low 7 hours, until chicken shreds easily.
Transfer chicken to a board, shred with two forks, then return it to the slow cooker.
Stir together cornstarch and water. Pour slurry into the pot and mix well.
Cut biscuits into 1-inch pieces. Toss with flour to prevent sticking.
Scatter biscuit pieces over the surface. Gently press to submerge in liquid.
Cover and cook on High 45–60 minutes, without lifting the lid, until dumplings are set.
Fold in peas and parsley. Cook 5 minutes more. Taste and adjust salt if needed.