Season chicken with basil, oregano, rosemary, salt, and pepper.

Melt butter in a large skillet over medium-high heat. Add chicken, skin-side down, and sear until golden brown, 2 to 3 minutes per side. Drain excess fat and set chicken aside.

Place potatoes in a 6-quart slow cooker. Stir in olive oil, garlic, and thyme, then season with salt and pepper. Add chicken in an even layer over top.

Cover and cook until potatoes are tender and chicken is no longer pink at the bone and the juices run clear, 7 to 8 hours on Low or 3 to 4 hours on High. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).

Serve immediately, sprinkled with Parmesan cheese and garnished with parsley.