Place the chicken thighs in the bottom of the slow cooker. Add the carrots, parsnips, potatoes, and onion on top.
Pour the chicken broth over the ingredients. Add the rosemary sprigs, thyme sprigs, half of the chopped parsley, salt, and black pepper.
Cover and cook on low for 6 hours until the chicken is tender and the vegetables are soft.
Remove the herb sprigs. Shred the chicken directly in the pot with two forks and stir to combine.
In a medium bowl, mix the flour, baking powder, and remaining salt. Stir in the milk, melted butter, and remaining parsley until a thick dough forms.
Drop spoonfuls of the dough onto the surface of the hot stew. Cover and cook on high for 1 more hour until the dumplings are cooked through and no longer doughy in the center.