Whisk the chicken broth and condensed soup in the slow cooker until smooth.
Stir in onion, celery, carrots, garlic, thyme, salt, pepper, and butter.
Nestle the chicken into the mixture, turning to coat. Cover and cook on Low for 5 to 6 hours until tender.
Shred the chicken in the cooker with two forks. Stir in the peas.
Toss biscuit pieces with flour to lightly coat, then scatter them over the top.
Switch to High, cover, and cook 60 to 90 minutes until dumplings are puffed and cooked through.
Rest 5 minutes. Sprinkle with parsley and serve warm.