I am not Snopes.com, but I and many others have looked into this claim, and we call FOUL. The so-called Pope Paul VI approval never happened. The closest thing was a scholarly article published in 1961 that proposed such the concession. Catholic Journalist John Allen finds implicit approval for this concession, by the fact that Pope Paul VI, when he became Pope, never explicitly rejected it.
John Allen, Catholic journalist?! A staff reporter is not a moral theologian, by training or temperament. In this case also not by intellect.
Regardless, of course Bl. Pope Paul “explicitly rejected it.” What on earth was Humanae Vitae?
Karol Wojtyla collaborated with the Pope in writing that Encyclical by the way, so there goes the theory being purported that TOB somehow contradicts it.