Is it permissible to use money taken from parents (such as an allowance or for a specific purpose like buying a computer) for expiation of an oath (kaffarat al-yamin), and is one who takes this money considered capable of feeding [the poor] or does he move on to fasting?

The allowance a father gives his son has two scenarios: either it is given for a specific purpose, in which case it must only be spent on that purpose, or it is given without specification, in which case the recipient owns it and may use it to pay for expiation.

Regarding the expiation for an oath (kaffarat al-yameen), the criterion for the obligation of feeding or clothing is that the person owns more than their basic needs at the time the expiation becomes due, whether this is from a gift or otherwise. So, if he owns what enables him to perform the expiation and it is surplus to his needs and the needs of those he supports, then fasting will not suffice for him. The father is not obligated to pay for his son's expiation, but if he volunteers to do so, it is good. The son is not required to save his allowance for expiation.