Tradition, drawing on the (non-canonical but very early) Gospel of James, gives us Saints Joachim and Anne as the names of the father and mother of the Mother of God. While not part of the Scriptural canon, there is no reason to doubt the historical veracity of the text (the normal scepticism of some modern commentators notwithstanding).
In the Protoevangelium of James, Joachim is described as a rich and pious man of the house of David who regularly gave to the poor and to the temple (synagogue) at Sepphoris. However, as his wife was barren, the high priest rejected Joachim and his sacrifice, as his wife's childlessness was interpreted as a sign of divine displeasure. Joachim consequently withdrew to the desert where he fasted and did penance for forty days. Angels then appeared to both Joachim and Anne to promise them a child. Joachim later returned to Jerusalem and embraced Anne at the city gate.
History of the feast
In the Roman calendar this is the feast of St Anne only, as the martyrology suggests:
"The departure from this life of St. Anne, mother of the Immaculate Virgin Mary, the Mother of God."
In the Benedictine calendar however both saints are celebrated on the same date.
The feast of St Joachim was added to the Roman Calendar in 1584, for celebration on March 20, the day after the feast day of Saint Joseph.
In 1738, it was transferred to the Sunday after the Octave of the Assumption of Mary. As part of his effort to allow the liturgy of Sundays to be celebrated, Pope Pius X transferred it to August 16, the day after the Assumption, so that Joachim may be remembered in the celebration of Mary's triumph. It was then celebrated as a Double of the 2nd Class, a rank that was changed in 1960 to that of 2nd Class Feast.
In the Roman Catholic calendar of saints (in 1969) it was joined to that of Anne, for celebration on July 26.
The Eastern Orthodox Churches and Greek Catholics commemorate Joachim on September 9, the Synaxis of Joachim and Anne, the day after the Nativity of the Theotokos.