It is customary for one to recite Kaddish as a source of merit for one’s deceased parent after the parent’s passing, even if the parent was exceedingly righteous. Hacham Ovadia Yosef explains in Halichot Olam (vol. 7, p. 508; listen to audio recording for precise citation) that although the parent certainly earned Gan Eden through his piety, nevertheless, the child’s recitation of Kaddish has the ability to elevate the parent’s soul to even greater heights in Gan Eden.
When Yahrtzeit is observed for a parent, the child begins reciting Kaddish at Arbit on Friday night before the day of the Yahrtzeit, through Minha on the day of the Yahrtzeit. Hacham Ovadia writes that if the Yahrtzeit falls on Shabbat, then one begins reciting Kaddish on Friday night the week before the Yahrtzeit, and thus ends up reciting Kaddish on two Shabbatot.
Hacham Ovadia notes that it is preferable when observing a Yahrtzeit to receive the Aliya of Maftir. Based on the Zohar, and the Ben Ish Hai (Toldot, Shana Sheniya, 21), he explains that the Aliya of Maftir brings greater satisfaction to the departed parent’s soul than other Aliyot, even the Aliya of "Mashlim," which includes a Kaddish recitation.
Hacham Ovadia in this context addresses the unusual situation of a father who does not want his son to recite Kaddish after the passing of the son’s mother, thinking that this might bring "bad luck" and cause the father to die. Of course, this is not true at all, but nevertheless, Hacham Ovadia rules that the son in this case is obligated to obey his father’s wish, and should not recite Kaddish. He should instead spend extra time learning Torah as a source of merit for his mother’s soul.
Summary: One recites Kaddish after a parent’s passing, Heaven forbid, even if the parent was very righteous, in order to elevate the parent’s soul to even greater heights. Each year after a parent’s passing, one recites Kaddish from the Friday night before the Yahrtzeit through Minha of the Yahrtzeit, and if the Yahrtzeit falls on Shabbat, then he begins Friday night the previous week. One observing a Yahrtzeit should try to receive the Aliya of Maftir. If one’s father asks him not to recite Kaddish after his mother’s passing, he should obey his father’s wish and spend extra time learning Torah in place of reciting Kaddish.
