ועל כל נפשת מת לא יבא לאביו ולאמו לא יטמא "He shall not go to any dead person nor defile himself for his father and mother" (Wayyigra 21:11). Unlike a regular Kohen, the Kohen Gadol (High Priest) is not allowed to attend the funeral of his immediate relatives, even his father and mother. The following parable, perhaps, best explains the reasoning.
There were three best friends who had to go separate ways to study for their careers. A few years later they met up together and told each other what they had learned. The first friend said that he had learned to create a telescope which could see from one end of the world to the other. The second said that he had learned to make planes. The third said that he had become a doctor.
The first one produced his telescope and saw that in a distant land, the daughter of the king was critically ill and about to die. "Quick", said the second, "let's get in my plane and get to them right away". As soon as they arrived the third began to heal her and snatched her from the jaws of death. The king was ecstatic and gave them a huge sum of money to be divided between the three of them. Not only that, but he gave them his daughter in marriage and that is where the problems started.
She couldn't marry all three of them so each one argued that she should marry him because without him they would never have succeeded. The king said that each one was equally important, so they should draw lots. His daughter, however, said the following: "It is true that up to this point, I would not have been saved without the intercedence of each of them, but going forward, I only require one of them. I am still not strong and require the doctor, so he is the one I am going to marry".
There are three partners in a person, his father, his mother and the Holy One blessed be He. After a certain point, however, the person is left with only One. It is G-d Who cares for him in all senses after the passing of his father and mother. The High Priest, who is on a higher spiritual level than all the Kohanim, must keep his spiritual purity intact for G-d's holy work.
(See Barukh Ta'am, Parashath Emor)
Baruch Ta'am, Rabbi Baruch Toledano, Hakham Barukh Toledano, Parsha,, Parasha, Emor, Kohen Gadol