In a leap year, when should one “celebrate” a yarzeit in ADAR 1 or ADAR 2 when a parent passed away in a non leap-year YEAR? Are their different opinions? What are they?
If the parent died in Nissan, then even if 12 months will be in Adar 2, the yahrtzeit is still during Nissan. If however the parent died during Adar of a regular year there are different opinions as when perform the various customs. The Shulchan Aruch (O:CH 568-7) and Sefardi minhag is to fast etc. during the second Adar. However the Rema, and the minhag of the Ashkenazim is to fast etc. in the first Adar. There is a third opinion, (Magen Avrohom, Gr”a) that essentially the child should fast and say kaddish during both Adars. The Mishna Berura (568-42) says that if someone comes to ask what he should do, it is correct to tell him to do like the opinion of the Magen Avrohom. Additionally if someone was mekabel on himself to fast on the day his parent died, he has to fast on both days, however if he wasn’t mekabel this on himself specifically, he should do according to the way he acted during the first time the yahrtziet was during a leap year.
Regarding fasting, nowadays many people don’t fast because of weakness, ( see Pischei teshuvos 568-7) especially if it will cause them not to be able to learn. However instead the child should learn and do whatever mitzvos he/she can that it should be a zechus for the parents neshoma.