Place the chicken thighs, Yukon potatoes, carrots, celery, onion, and garlic into the slow cooker.
Pour in the chicken broth, then add the thyme, salt, and black pepper. Stir once to combine.
Cover and cook on low for 6 hours until the chicken is tender and the vegetables are soft.
Stir in the heavy cream and Parmesan cheese until the cheese melts into the broth.
Cover and cook for 20 more minutes on low to warm everything through.
Ladle into bowls and sprinkle with fresh parsley before serving.