Place chicken pieces in a large Dutch oven; add enough cold water to cover the chicken. Add carrots, celery, onion, salt, thyme, rosemary and pepper. Bring to boil; then reduce the heat to medium-low or low, so that the liquid is at a simmer.
Cover the pot and simmer gently for 1 hour. Remove the chicken from the broth. When the meat is cool enough to handle, discard the skin and bones. Shred the meat into bite-size pieces. Set aside.
Taste the broth and season with additional salt and pepper, if necessary. Bring the broth to a boil over high heat.
While the broth comes to a boil, prepare the dumplings. Place self-rising flour in a large bowl. Add the butter and use a pastry blender (or your fingers) to cut the butter into the flour until the mixture resembles coarse meal. Add the buttermilk and parsley; stir with a fork until the dough comes together.
Turn the dough onto a floured surface. Knead 4-5 times, then pat to ½ inch thickness. Pinch off dough in 1 ½ inch pieces, dropping the dough pieces into the boiling broth. Reduce heat to medium-low so that the broth is at a gently simmer. Let the dumplings cook for a few minutes undisturbed, then gently turn them with a fork or spoon so that they cook evenly. Continue simmering until the dumplings are cooked through, about 10 minutes total.
Stir in the chicken.
Ladle the soup into bowls; garnish with additional fresh parsley.