> What in the parable suggests the man is inviting the demon back in?
The context tells us this. It's the lack of repentance that functions as an invitation. The men of Nineveh will be a witness against these evil people because the men of Nineveh had the good sense to repent (12.41). Even a "pagan," the Queen of the South, had the good sense to recognize when she was in the presence of wisdom, but these religious leaders don't, and they open themselves up to greater deceit. If there's a spiritual void in your life (12.44), if you don't fill it with something true and wise, it will be filled by another with deceit and foolishness.
> If that is the main point, Jesus doesn't make it explicit. Jesus says this will happen to the current generation but he doesn't really say why.
He implies why. (1) They fail to repent. (2) They fail to recognize wisdom when it's right in front of their eyes. (3) They fail to turn their spiritual attention to truth.
> The reason the spirit returns is because it does not find rest in the dry places.
It's a parable, not an allegory. In an allegory, every piece has a metaphoric meaning. In a parable, what counts is the point (the moral of the story), not the details.
But still, why would Jesus choose this image of the "dry places"? In the ancient world, the wilderness (along with the sea and darkness, et al.) were considered to be places of non-order: chaos. Therefore a place like the desert was considered to be "the haunt of demons," and was populated by other elements of non-order, such as jackals and scorpions. It lacked life-giving water. It was a place of danger.
In making this analogy, Jesus is saying that this man in the parable is a more suitable dwelling place for the demon, he has made his life so fitting for the demon's occupation, even more fitting than the desert itself! As the WWI song goes, why would you live on the farm after you seen Paris? (
https://en.wikipedia.org/wiki/How_Ya_Gonna_Keep_%27em_Down_on_the_Farm_(After_They%27ve_Seen_Paree)%3F)
Jesus's point is that this person has allowed their life to be the perfect and comfortable living quarters for this demon and his buds. And He likens that to the religious leaders He's talking to.