A dad joke is any joke that’s so terrible it elicits groans and ruins lives. It need not originate from a dad, per se, but the common cultural understanding is that most of them do. You’ll find them in wedding speeches, rogue Facebook comments, and conversations in line at the hardware store. From there they pervade the very fabric of our society so that even the best of us, dad or not, feel compelled to hit every "I’m bored" with a "Hi bored, I’m Dad."
Shakespeare was the father of English literature, but he was also the father of three actual kids, so it’s no wonder his plays are full of puns and anti-jokes so objectively corny you can practically hear sixteenth-century audiences shouting, "COME ON, WILL." These were his worst (and by worst I mean best) offenses.