"One of them's father is a prison guard." is not correct. There's no such word as "them's." "The father of one of them is a prison guard." is fine. You could also say, "One of their fathers is a prison guard."
If you want to use an apostrophe, then "them" has to be specified, so you can make it into a possessive form. For example, the following are correct for various kinds of "them