When one concludes the twelve-month mourning period following a parent's passing, an event called an "Arayat" is customarily held. The word "Arayat" means "reading," and the event is so named because portions of Tehillim and of certain Kabbalistic works are read. Food is normally served as the event.
Hacham Ovadia Yosef ruled that if a person wishes to continue reciting Kaddish even after the twelve-month mourning period, he may do so, and this will bring benefit to the deceased's soul. Although the standard punishment in Gehinam lasts for only twelve months, Rav Haim Vital (1543-1620) writes that some wicked people are not even ready to be brought to Gehinam immediately after their death, and thus their period of suffering in Gehinam – which could last up to twelve months – might begin only some time after death. Therefore, there is benefit to the Kaddish recitation even beyond the twelve-month mourning period. In fact, the Hesed La'alafim (Rav Eliezer Papo, 1785-1828) instructed his sone to recite Kaddish after his passing at least once a day for the rest of their lives.
Nevertheless, this is not the commonly accepted practice. Generally speaking, people recite Kaddish for their parents after the first twelve months only following a Torah class or Tehillim reading, but not during the prayer service. Sometimes, a Rabbi might advise a congregant to continue reciting Kaddish after the first year of mourning in order to encourage the congregant to continue attending the prayers.
Summary: Although there is benefit to reciting Kaddish for a deceased parent even after the twelve-month mourning period, it is customary not to, except after a Torah class or Tehillim reading.