In listing the relatives which a kohen can/should (machlokes Rishonim whether it is a chiyuv or a reshus) become tamei to bury, the Torah firsts lists his mother and then father. Yet when it comes to listing who the kohen gadol may not become tamei for and who the nazir may not become tamei for the Torah places the father before the mother. Why here does the Torah place the mother first?
The Ibn Ezra gives what I would call an actuarial answer: men outlive women. Women, in this case the mother, are listed first because they die first. I believe the modern metziyus is that women in fact live longer, but maybe that was not true in the past given the higher chance of death in childbirth.
Meshech Chochma points to the Rambam's view (Aveil 2:7) that m'd'oraysa a husband may not become tamei to bury his wife, but the Chachamim allowed it. How can they uproot a doraysa prohibition? Rambam answers that the Chachamim treated the wife as a meis mitzvah. A meis mitzvah means there is no one responsible for burial. When it comes to a wife, the Chachamim defined the husband as the only one with a close enough relationship to feel responsible to bury her.
The Torah lists mother first in the pasuk because one might have thought that if a person's father is alive, the father is responsible for the mother's burial and the kohen son should not make himself tamei, kah mashma lan that he can, and according to the Rambam, must.
The Toras Kohanim seems to say that the pasuk lists the mother first and father second because while we know exactly who a person's mother is, absent a DNA or paternity test we only know who a father is based on rov/chazakah. If two people are married, we assume that the father is the mother's husband, but an assumption cannot outweigh certainty.
Aside from the technical detail that this is not much of a chiddush (beis din can administer the death penalty based on rov and chazakah, so it surely is a good indicator or paternity), the Rishonim and Achronim ask an even more basic question on the Toras Kohanim: any doubt as to the paternity of the "kohen" is MORE of a reason to allow him license to become tamei, not less! If the father in question is not the true father of this individual, then odds are that he is not an kohen and there is no issur in him becoming tamei in the first place.
In other words, m'mah nafshach: either this is the son of a kohen and he can become tamei for his father, or he is not the son of a kohen and there is no issur on him becoming tamei. What hava amina can there be to say otherwise that would cause us to think there is some chiddush here?
So hopefully I will have more time to write and get to an amazing yesod of R' Yaakov Kaminetzky's. Stay tuned, and see the Malbim as well.