Please list the order of priority for a mourner in leading the davening. For parents, siblings etc. Why does a mourner for a brother in Shloshim not have priority over a child mourning for a parent after Shloshim?

Meriting a Neshama is done by the son. Therefore, he has priority, all other relations do not have any priority. A short list of the priorities for leading the prayer: Son in Shiva'a prior to son in Shloshim, Son in Shloshiom prior to son in year and Yartziet, Yartzeit prior to son in year. For explanation at length look in Shulchan Aruch Yoreh Deah 376:4 the Rm"a, and in Orach Chayim 132 and the Be'ur Halacha there. In any case if the minhag of the town or community is different one follows that minhag and one should not quarrel as the peace and harmony profits more for the Neshama of the Niftar. Rabbi Ro'i Margalit