It could be argued that sending unsolicited mail to a private residence wherein minor children could presumably reside made for a situation wherein children could be exposed to pornography. Children in homes are a common occurrence. Unsolicited gay porn in the mail is not.
Had the mail been solicited the responsibility would be on the party placing the order.
The obvious counter argument is the children should be monitored when using the computer and the fault is with the father for being a bad and lazy parent. That is one of the problems with a suit like this -- every aspect of the family's parenting is open for discovery. If I was defending this (I am not a lawyer, but I did stay at a Days Inn last week), I would make the father out to be a lax parent who lets his kids do anything they want and doesn't monitor their activities on the Internet or in RL.
This could get ugly and, as I said, the target can just go to the Barbados and start the same site with a different name if things look like they will go south. It is a civil trial so you can't detain people, nor can you freeze assets before a judgment is rendered.
Loss and loss of face and reputation, worst case. Win a Pyhrric victory and still lose face and reputation, best case.
And before you all give me the "this must be STOPPED" speech, the bottom line is that 80% of all email is spam, most of which is sexual in nature. If Daddy didn't teach his kid safe computing (never open an email from anyone you don't know), then he is part of the problem. It could have been a virus that wiped out the whole computer, so I don't think I am off base here.