After the Shir Shel Yom, the custom is to recite Kaddish Yehe Shelama, although the Ashkenazim recite it after Alenu Leshabe’ah. The Rama (Orah Haim 132:2) notes that this Kaddish is also called Kaddish Yatom (orphan’s Kaddish) and is normally recited by one who has lost a parent, but may also be recited by one whose parents are both alive and do not object to him reciting it.
There are situations in which, fortunately, every congregant’s parents are alive and the rabbis discuss whether or not Kaddish Yehe Shelama need me recited at all. According to the Arizal (Sha’ar HaKavanot) the Shaharit prayer has seven Kaddishim, which act as separators between the different parts of the prayer. Each part of the prayer represents a different spiritual world and the Kaddish helps transition between those worlds. It follows that Kaddish is not just beneficial for the elevation of the souls of the departed, but is also an essential component of the prayer, and therefore even if no one has lost a parent, it should still be recited. Rabbi Ya’akov Moelin (Shu”t Maharil, § 64) declares emphatically that Kaddish Yehe Shelama should still be recited even if none of the congregants have lost a parent. Furthermore, Rabbi Ben Zion Uziel (Shu”t Mishpete Uziel, Orah Haim § 2) writes that the Kaddish should not be ascribed auspicious powers, but rather is a fundamental part of the prayer and should be recited at all times. However, Rabbi Eliezer Deutsch (Shu”t Pri HaSade, vol. 4, § 92) writes that although strictly speaking the Kaddish should be said, if one’s parents will object or if it will be looked upon oddly in the synagogue, the custom is to omit Kaddish. It appears that this common practice is partially because of the superstitious belief that it is inauspicious for someone who has both parents to recite it, although this is incorrect according to the letter of the law.
Summary: Strictly speaking, Kaddish Yehe Shelama (aka Kaddish Yatom), should be recited at all times, regardless of whether one of the congregants has lost a parent.