It is indeed possible to switch back and forth between households in the same town, I often do that. It's very easy, just go to Edit town and click on the circular button with the 2 little houses on the left side of screen. But before you do anything, I strongly recommend that you backup your Saves folder, so if anything goes wrong you could always go back to your old save file.
If you want to be absolutely sure that the parents are going to stay in the neighborhood, it's best to have Story progression turned off. Otherwise there might be a chance that they are "moved out of town", that is deleted, and you wouldn't want that to happen, would you?