If a mourner is reciting Pesukeh De'zimra when the congregation reaches the point where the mourners recite Kaddish – just before Alenu, at the end of the service – is the mourner allowed to interrupt his Pesukeh De'zimra recitation to say Kaddish? As a rule, it is forbidden to interrupt Pesukeh De'zimra, and the question thus arises whether reciting Kaddish for a parent qualifies as an exception where interrupting Pesukeh De'zimra is justified.
The Mishna and Gemara in Masechet Berachot address the situation where a person is reciting Shema, or the blessings before or after Shema, when his father passes by. Strictly speaking, Halacha permits the person to interrupt his recitation for the sake of extending a greeting his father. As a practical matter, this Halacha does not apply nowadays, because protocols for greeting parents are not as formal as they were in the past, and thus it is not necessary to interrupt one's prayer in order to extend a greeting to a parent. However, the concept underlying this Halacha is relevant with regard to the Kaddish recitation. Reciting Kaddish for a parent fulfills the Misva of Kibbud Ab Va'em (honoring parents), and we therefore apply the rule that was observed in the past regarding respectful greetings to one's parent during the prayer service. The benefit brought to the deceased parent's soul through the recitation of Kaddish indeed justifies interrupting Pesukeh De'zimra, and thus the person in this case should recite the mourners' Kaddish together with the congregation.
If the person sees that the congregation will soon arrive at the point of the mourners' Kaddish, he should, if possible, complete the Mizmor (chapter) that he is currently reciting, as it is preferable to make an interruption in between two Mizmorim instead of interrupting a Mizmor.
Summary: If a mourner is reciting Pesukeh De'zimra when the congregation reaches the point of the mourners' Kaddish, he should interrupt Pesukeh De'zimra to recite Kaddish. Preferably, he should try to complete the Mizmor (chapter) that he currently recites before Kaddish need to be recited, so he does not interrupt a Mizmor.