I don't really think the best solution for a man-child who cannot do housework to save his relationship, is for him to move back home with his parents. You do realize that you're not actually taking out the metaphorical trash, you're just kicking it under the rug, right?

Work out a schedule for housework, and make him stick to it. Explain to him that if he doesn't, you will stop doing shit for him - wash only your clothes, cook your own meals, shop for one. He can take care of himself, he's a grown man, you're not his mother. He then has a choice - shit or get off the pot. Set a timeline - give him a month/two and evaluate then. But don't think for a second that if you make him move back to his parents the situation will magically change when at some point in the future you move back together.