Greeting a Mourner After Thirty Days

Is there a source that permits greeting a mourner after thirty days within twelve months for a father and mother?

According to the basic law, one should not greet a mourner, as he is not in peace for twelve months. However, one who is lenient in this has upon whom to rely. And if the lenient person greets the mourner, he must respond.

Shulchan Aruch, Siman 385, Se'if 1: "From seven to thirty days, one asks about the peace of others who are in peace, but others do not ask about his peace, and certainly, he responds to one who asks about his peace. After thirty days, he is like any other person. This applies to other relatives, but for a father and mother, one asks about the peace of others after seven days, and others do not ask about his peace until after twelve months." The Rema writes: "There are those who are lenient nowadays in the matter of greeting a mourner after thirty days, and they have no reason, unless to say that what we practice is not called greeting as in their days." In explaining the custom, we find three opinions among the poskim: the opinion of the Taz Orach Chaim, Siman 396, Se'if 3, who hold that there is no law of greeting a mourner after thirty days for a father and mother. The Magen Avraham, Siman 594, Se'if 21, writes that the custom to be lenient is because they thought it permissible to say "Shalom Aleichem," and the entire prohibition is in greeting, and so it seems in the Shach, Siman 385. In the Gilyon Maharsha, it is written that they hold the prohibition is only in "Shalom," which is the name of the Almighty. The Ran already wrote in Moed Katan, page 21b: "But for a father and mother, one does not ask about his peace until after twelve months. But he asks about the peace of others after seven days, even if his father and mother died. In our generations, they do not observe this, and the Lord in His mercy will forgive our sins." The Taz writes that the custom is not to prohibit greeting after thirty days, and so writes the Pri Megadim, Mishbetzot Zahav, Siman 396, Se'if 3, and so it seems in the Kitzur Shulchan Aruch, Siman 290, which omits the law of twelve months for a father and mother. And so writes the Aruch HaShulchan, Yoreh De'ah, Siman 385, Se'if 2: "And it seems to me that now even within thirty days they do not observe this, and perhaps for this reason." There are those who write that since today many do not observe this custom, it is considered as forgiveness, as explained in the Shach, Siman 344, Se'if 9, that a father can forgive his son for the twelve-month mourning. However, from the simple language of the Shulchan Aruch and the Rema, it seems they rejected this custom and hold that one should not ask about the peace of the mourner for twelve months. And so instructed our teacher, the Gaon, Rabbi Avraham Fried, shlita.