Is it permissible to give zakat to one's parents to enable them to perform Hajj or Umrah, or to purchase a durable good that would alleviate the burdens of life for them in their old age?

It is not permissible to give zakat to one's parents. Nor is zakat to be spent on performing Hajj or Umrah for anyone, because it is the right of the poor, the needy, and those mentioned with them in the verse: "Zakat expenditures are only for the poor and for the needy and for those employed to collect [zakat] and for bringing hearts together [for Islam] and for freeing captives [or slaves] and for those in debt and for the cause of Allah and for the [stranded] traveler - an obligation [imposed] by Allah. And Allah is Knowing and Wise." (At-Tawbah: 60). The eight categories mentioned in the verse are the rightful recipients of zakat.