The riddle
When Does a Joke Become a Dad Joke?
Apparent and a parent
Said quickly, "becomes apparent" sounds like "becomes a parent". Careful speakers may keep the vowels in the middle apart, but in ordinary talk the two phrases blur together. The riddle's claim is then easy to read: a joke turns into a dad joke at the moment someone becomes a parent. Wiktionary lists apparent as clear or plain to the understanding, so the answer is also true in the ordinary sense of the word.
Where the term dad joke comes from
Wikipedia says the exact origin of the name is unknown, but dad jokes were published under that name at least as early as 1987. The first printed use it credits is a June 1987 column by Jim Kalbaugh in The Gettysburg Times. It describes a dad joke as usually a pun, often a one-liner or a question and answer, generally inoffensive and told sincerely or to get a groan.
The pun in 1890 and 1903
The pun on apparent is far older than the term. Eaton's 1890 compilation asks why a clearly discovered riddle is like a letter written by a child to its mother, and answers, "It is too apparent (to a parent)." Rivers's 1903 book asks, "When was wit a father?" and answers, "When a pun became apparent (a parent)." The 1890 printing comes 97 years before that 1987 column.
Why the riddle fits itself
The riddle is itself a pun told as a question and answer, so by Wikipedia's description it is a dad joke. The same article quotes a 2021 Men's Health version, which shows the riddle still circulating. Tell it with a deadpan voice and watch who groans first. A close cousin that turns on the sound of a name: what do you call a potato who reads the news.