If one has a parent that is within the 12 months of mourning of his parent, can one give the parent a birthday gift of any sort? If not a present, can one bake them a cake or special dish if they request?
After the thirty days (sloshim) it is permitted to buy a modest present for a parent, and this is certainly true where the parent will take offence if a gift it not given.
The problem of giving a gift is defined as she’elas shalom – a gift is not less than giving Shalom to another (see Maharil no. 31; Rema, Yoreh De’ah 385:3).
The room for leniency is that the Taz (Orach Chaim 696:3) writes that the custom today is not to be particular about she’elas shalom after thirty days, and that it it is therefore also be permitted to give gifts. [The ruling is not entirely clear, and some have asked (see Eliyah Rabbah) that the Rema never states that today’s custom is to be lenient concerning she’elas shalom, but only that the custom in some places is to be lenient.]
The Mishnah Berurah (21) quotes this ruling, though he brings it from Magen Avraham (who explains differently), which makes his opinion a little unclear (see Divrei Sofrim, aveilus, Eimek Davar 47); he adds, however, that the gift should not be something that causes simchah. The same ruling is given by the Derech Chaim.
Based on the principle that the halachah follows the lenient opinion in aveilus (seeShiurei Berachah 374, who writes that he has a mesores from all morei horaa that this applies even to disputes between poskim), and in view of the need involved, it would be permitted to mark birthdays with a modest present — not perhaps as grand as those of other years, which would be a “gift of simchah,” but something nonetheless.
Best wishes as always.