The Gemara in Masechet Erubin (40) addresses the question of whether we must make mention of Rosh Hodesh in the Rosh Hashanah prayer. Rosh Hashanah, of course, is observed on the first day of Tishri, and is thus also Rosh Hodesh, and in the times of the Bet Ha’mikdash the usual Rosh Hodesh sacrifices were brought in addition to the special Rosh Hashanah sacrifices. We might therefore assume that we should mention the occasion of Rosh Hodesh in our Rosh Hashanah prayers, just as we mention it on any ordinary Rosh Hodesh. The Gemara concludes, however, that we do not mention Rosh Hodesh in our Rosh Hashanah prayers, because "Zikaron Ehad Oleh Le’kan U’le’kan." This means that the mention of "Yom Ha’zikaron" in reference to Rosh Hashanah incorporates the occasion of Rosh Hodesh, and therefore no special mention of Rosh Hodesh is required. This is codified in the Shulhan Aruch, and this is the widespread practice.
Interestingly, we find this concept of "Zikaron Ehad Oleh Le’kan U’le’kan" applied in other halachic contexts, as well. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in his work Torah Lishmah (412), addresses the case of a man whose father passed away, and within twelve months of his father’s death his mother passed away. The man posed the question of whether the Kaddishim he recited each day for his father were able to cover his obligation to his mother, as well, or if perhaps he needed to hire somebody to recite Kaddish for his mother. The Ben Ish Hai ruled that the man’s Kaddish recitation covered both deceased parents, and in this context he invoked the Gemara’s ruling of "Zikaron Ehad Oleh Le’kan U’le’kan." He concedes that the Gemara’s ruling does not necessarily provide proof, as the Gemara inferred from Pesukim that the term "Yom Ha’zikaron" can refer to Rosh Hodesh. Nevertheless, this is a similar concept where one recitation covers two different requirements. The Ben Ish Hai draws further proof from the Shulhan Aruch’s ruling that if a person accepts upon himself a series of fasts, and a public fast day – such as Shiba Assar Be’Tammuz – falls on one of those days, he does not have to fast an additional day, as his fast on that day counts toward both his obligation of Shiba Assar Be’Tammuz and his voluntary series of fasts. By the same token, the Ben Ish Hai writes, a single Kaddish recitation can count for both parents. The Ben Ish Hai adds that to the contrary, it would be disrespectful to hire somebody to recite the Kaddish, as a son’s personal recitation of Kaddish for a parent is especially meaningful and valuable.
Another application of this concept is the question addressed by the Poskim of whether a person who is hired to recite Kaddish may receive payment from several people to recite Kaddish on behalf of their deceased family members. One might argue that once he is reciting Kaddish for one person, he may not accept payment for his Kaddish to count for others. Rav Moshe Feinstein (Russia-New York, 1895-1986), in his Iggerot Moshe (Yoreh De’a 254), writes that the person must recite at least one Kaddish per person per day, and he must explicitly notify all those paying him that this is the arrangement. Hacham Ovadia Yosef, however, in Yabia Omer (vol. 8, Yoreh De’a, Siman 37), rules that we may apply to this case the concept of "Zikaron Ehad Oleh Le’kan U’le’kan," such that one recitation of Kaddish can count for multiple people. Hence, one may receive payment from several individuals to recite Kaddish on behalf of their deceased family members, as his Kaddish recitation can indeed apply simultaneously to several different people. This ruling is codified by Rav Shemuel Pinhasi in his work Haim Va’hesed, where he notes that this was also the position of the Sedeh Hemed (Abelut, 154) and the Sitz Eliezer (7:49).
Summary: Even though Rosh Hashanah is also Rosh Hodesh, we do not mention Rosh Hodesh in our prayers because the phrase "Yom Ha’zikaron" refers to both Rosh Hashanah and Rosh Hodesh. One whose parent dies during the period he is saying Kaddish for the other does not have to hire somebody to recite Kaddish, as his Kaddish recitation may count for both parents. Similarly, one who is paid to recite Kaddish may accept payment from several people, and his Kaddish recitation counts for all of them.