Dad the sad dog is a character from the book Hop on Pop. He is the father of three puppies.
In his chapter of the book, he is visibly upset about the day that he has had. It is never stated what specifically has made him so sad, but given the fact that's he's wearing a tie, we can infer it has something to do with his job in corporate America. His three puppies are very concerned about their father's mental state, and appear to have agreed not to provoke him further.