For Corned Beef:
Put the carrots and onions on the bottom of slow cooker, and put the corned beef on top. Arrange the chopped cabbage around the beef. Add the apple juice and 1 cup of water along with the contents of the spice packet. Cook on low 6 to 8 hours until the beef is tender. Remove the beef and vegetables to a platter and keep warm.
Bring a large pot of salted water to a boil and add the potatoes. Cook until the potatoes are tender, about 12 to 15 minutes. Drain and return them to the pot.
Add the butter, garlic, parsley, and salt, and pepper, to taste. Gently combine so that all the potatoes are evenly coated. Transfer to a serving bowl and serve with corn beef and vegetables from the slow cooker.