Is it permissible for a son to atone for his father by buying and then manumitting a slave, and is it permissible for him to atone for his mother?

Expiations (kaffarat) are performed by feeding or freeing slaves on behalf of those for whom the expiation is due, in two ways:

1. With the permission and request of the person for whom the expiation is due: It is valid and sufficient for anyone to perform it on their behalf. The proof for this is the story of the man who had intercourse with his wife during Ramadan, where the Prophet (peace be upon him) gave him food to give in charity. 2. Without their permission or knowledge: What is fed or freed on their behalf is not sufficient. This is the view of the Shafi'is, Hanbalis, and Qadi Abu Yusuf, because worship is not valid without intention (niyyah), and expiations have an aspect of worship and drawing closer to Allah Almighty, and are not merely a financial fine.

In conclusion, it is permissible for a child to free a slave or feed the poor on behalf of their parents if they permit it, and the child will be rewarded for their dutifulness.