Place the potatoes into a large pot and cover with cold water. Salt the water with 1 teaspoon of salt. Cover the pot and bring to a boil over high heat. Reduce heat to simmer and cook for 15 minutes. Add the green beans to the pot and simmer five minutes more. Drain. Return the potatoes and green beans back to the pot. Add the butter, garlic, parsley, lemon zest, and a generous amount of salt and pepper. Stir until the butter is melted and everything is evenly coated. Taste for seasoning and add more salt and pepper as needed. Keep the potatoes and green beans covered and warm until ready to serve, giving a quick stir before plating.