The Gemara teaches, "Mechabedo Be’hayav Mechabedo Be’moto" – one must give respect to his parents both during their lifetimes and after their passing. There is a disagreement as to whether the requirement to honor parents after their passing is part of the Torah obligation of honoring parents, or a Rabbinic enactment. Clearly, however, one is obligated to continue respecting his parents after they have passed on. This is done in several different ways, such as by lighting a candle in their memory on their Yahrtzeit, performing Misvot in their merit, and reciting Kaddish during the week of the Yahrtzeit.
The Shulhan Aruch specifies one particular manner of respecting parents after their passing, namely, adding the words "Hareni Kaparat Mishkabo" after mentioning the parent’s name during the first year after his passing. If one cites a Torah thought or a teaching in his father’s name, he should say these words, which mean, "I am hereby an atonement for his rest." This means, literally, that the son accepts upon himself whatever punishment the father might deserve to endure in the afterlife. The Poskim point out that this is not really what the child means, because although a child can help bring his parent to the world to come, and save him from Gehinam, it is questionable whether he can take upon himself punishments that the father was to receive. What this means is that the child would be prepared to accept the punishments in his father’s place. This is a very meaningful expression of honor, as the son expresses his wish that he could receive whichever punishments the father might deserve in his place.
"Hareni Kaparat Mishkabo" is said only when one is quoting his father as teaching something, such as a halachic ruling or an insight into proper conduct. When mentioning one’s father in other contexts, this is not required. The Rama (Rav Moshe Isserles of Cracow, 1530-1572) adds that this should be said after mentioning one’s mother’s name, as well, in the feminine form – "Hareni Kaparat Mishkabah." Although one does not normally quote Torah teachings from his mother, this is relevant if one mentions his mother’s religious customs and practices, such as when lighting Shabbat candles, separating Hallah, and so on.
The requirement to say, "Hareni Kaparat Mishkabo" applies for the entire twelve-month period after a parent’s passing. As opposed to the Kaddish recitation, which ends after eleven months, saying, "Hareni Kaparat Mishkabo" continues during the twelfth month.
After twelve months, when one mentions his father he should say, "Zichrono Li’bracha" or "Alav Ha’shalom."
Summary: The obligation to honor one’s parents continues after a parent’s passing, and is fulfilled then through the various customs related to the Yahrtzeit, and performing Misvot in the parent’s merit. Additionally, for twelve months after a parent’s death, whenever one cites a teaching or insight in one’s parent’s name, he should say after the name, "Hareni Kaparat Mishkabo."