Place the chicken breasts in the bottom of the slow cooker. Arrange the quartered potatoes and trimmed green beans around and on top of the chicken.
In a small bowl, whisk together the lemon juice, lemon zest, minced garlic, olive oil, dried oregano, salt, black pepper, and chicken broth.
Pour the lemon mixture evenly over the chicken, potatoes, and green beans.
Cover and cook on low for 6 to 7 hours, until the chicken reaches an internal temperature of 165°F and the potatoes are fork-tender.
Sprinkle the chopped parsley over the finished dish before serving.