A long time ago, a woman who worked for me asked how Catholics know the name of Our Lady's parents, asserting that there is nothing in the Bible to support such a claim. She was a member of an Assemblies of God church, although she had been baptized a Catholic. She also liked to accuse Catholics of idolatry, and make remarks about the pope being the Antichrist, and so on. That day she wasn't interested in St. Anne or inquiring about the faith.
I replied to her challenge, "I don't know for sure."
"So why do you accept it as a fact?"
I smiled and shrugged, and said, "Tradition?"
She smirked and went away triumphant - satisfied she won a battle against a Catholic - proving how dumb I was.
I continued my work in peace.
Evangelization isn't about winning debates and arguments. Sometimes it is better to lose.
Art: St.s Joachim and Anna at the Golden Gate, Giotto, Scrovegni Chapel, Padua. Incidentally, the names attributed to Our Lady's parents is part of Sacred Tradition, arising from early apocryphal texts. For a quick history of the saints click here.
Bonus Factoid: The Blessed Virgin was conceived naturally by her parents, though preserved from Original Sin in her conception, hence her name: Immaculate Conception.
"We declare, pronounce, and define that the doctrine which holds that the most Blessed Virgin Mary, in the first instance of her conception, by a singular grace and privilege granted by Almighty God, in view of the merits of Jesus Christ, the Savior of the human race, was preserved free from all stain of original sin, is a doctrine revealed by God and therefore to be believed firmly and constantly by all the faithful." - Ineffabilis Deus