The Poskim addressed the situation of a man who, after his wife's passing, asks his son not to recite Kaddish for her. He might feel uneasy about his son saying Kaddish because people who see this might mistakenly think that he, the father, had passed away, or might just ask questions which would make the father uncomfortable. Should the son obey his father's wishes, or should he recite Kaddish in defiance of his father to honor his mother's memory?
The preferred solution in this case is for the son to ask a Rabbi or someone else whom the father respects to speak to the father and try to change his mind. The anxiety that some people feel when it comes to Kaddish, while understandable, is unnecessary. There is no risk whatsoever to a person if his child recites Kaddish, and there is no reason for concern. However, if the father's mind cannot be changed, and he insists that his son not recite Kaddish for his mother, then the son should obey his father's wishes and ask somebody else to recite Kaddish for his mother.