Is a person required to be Noheg Aveilus for a parent who is not frum? If yes, is there a difference if they went off the Derech and can’t be considered a Tinok Shenishbah?

Shalom, Thank you for your question. The Shulchan Aruch (Yoreh Deah, 345,5) writes that those Jews who completely separated themselves from the Jewish people, and do not include themselves in the Jewish community are not mourned for when they die. Exactly what is considered as having separated from the community is a difficult question to answer, especially in today's day and age when the majority of the Jewish people do not (unfortunately) follow halacha or even keep Shabbat. It would be difficult to say that every non religious Jew does not have to be mourned for, when so very many of them truly carry the name of Israel with pride, and do wonderful things for Am Yisael. Also, so very many of them have never had the opportunity to learn or get close to Torah and Mitzvot. With that in mind, one should not rush to decide that a particular person is outside the fold. On the other hand, if they actively worked against the Jewish people (such as missionaries etc), then there is room to say that one should not sit in mourning for them. The Rema writes (Y.D. 374,6) that it is permitted to mourn for people even if one is not obligated. Because of this, in cases of doubt one should mourn the dead, as there is certainly no sin involved by doing so, and in most cases it brings only good and harmony to the family. This is especially true with those who were not bought up religiously (the Tinok ShNishbah you mention), but is also true in many cases where the person went “off the derech” (as you put it) – because this so often happens because of emotional difficulties and not because of true rebellion against Hashem. Perhaps the best advice would be to examine each case on a individual basis – but in general the practice is to mourn for all Jews, despite their not being religious. May you be blessed with only good tidings and ling life for all.