There is is a nice summation of the Laws of Kaddish on Chabad.org Paragraphs 3–4 seem relevant to your question (emphasis added):
The period that the mourner recites the Kaddish for parents is, theoretically, a full calendar year. The deceased is considered to be under Divine judgment for that period.
Some communities, therefore, adhere to the custom that Kaddish be recited for 12 months in all cases. However, because the full year is considered to be the duration of judgment for the wicked, and we presume that our parents do not fall into that category, the practice in most communities is to recite the Kaddish for only 11 months. Even on leap years, which last thirteen months, the Kaddish is recited for only 11 months. We subtract one day, so that we terminate the Kaddish in time to allow a full 30 days before the end of the 12-month period. Thus, if we begin on the eighth day of the Hebrew month of Cheshvan, we end on the seventh day of Tishre.
If a parent insists on the child's reciting Kaddish for the full 12 months, there is surely no reason not to obey him. If children feel this might bring public dishonor to their parent, they should recite only the Rabbi's Kaddish in the twelfth month, despite his request. This is a practice worthwhile encouraging in every case for all parents.