Is it permissible to give charity from one's private funds to the children of a deceased sister, and would this be considered ongoing charity (sadaqah jariyah) on behalf of one's parents?

May Allah reward you for your kindness to these two individuals, as it is an act of righteousness and maintaining kinship ties. Your charity to them is better if they are in need of money, for it combines charity and kinship. Your intention to give charity on behalf of your parents is permissible, and they will benefit from it.