Children cannot make up fasts on behalf of their living parents
However, if the parents are still alive, children are not permitted to fast on their behalf, according to an article published by Maktabah Albakri.

FOR those who still have outstanding fasts from last Ramadan, it is encouraged to make them up as soon as possible before the blessed month returns.
An important aspect to consider is the fasting obligations of our late parents or guardians.
It is not uncommon for children to observe fasts on behalf of their deceased parents.
However, if the parents are still alive, children are not permitted to fast on their behalf, according to an article published by Maktabah Albakri. Fasting on behalf of a guardian is only allowed after their passing.
This fact is further supported by the view of Imam An-Nawawi, who stated: "In this matter, Imam Asy-Syafie has two well-known opinions. The first and most widely accepted opinion is that fasting on behalf of a deceased person is not permissible and such a fast is not valid in principle.
"The second opinion is that it is recommended (mustahab) for the deceased’s guardian to fast on their behalf. This fast is valid and will atone for the deceased’s sins, making it unnecessary to offer fidyah (compensation) to others.
"This second opinion is the sound one and has been endorsed by our scholars among the experts in fiqh and hadith, as evident from authentic and clear hadiths on this matter," he said, as quoted in Article #1740: Making up fasts for a living father.

This is supported by a hadith narrated from Aisyah RA, in which Prophet Muhammad SAW said: "If a person dies and has outstanding fasts, then his heir should fast on his behalf." (Hadith narrated by Al-Bukhari and Muslim).
"Based on this hadith, it is clear that only those who have passed away need their fasts to be made up by their heirs. Meanwhile, for those who are still alive but unable to fast due to a valid reason (uzur syarie) and will never be able to make it up in their lifetime, they must compensate by paying fidyah," the article stated.
Regarding fidyah, Maktabah Albakri explained that it is best for a person to first attempt to make up their missed fasts on as many days as possible.
“This approach is more ihtiyati (cautious) in religious matters compared to immediately opting for fidyah and neglecting the obligation of qada (making up missed fasts)," the article stated.
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