Short answer
With honour and patience. Among the deeds God (﷾) loves most, the Prophet Muhammad (ﷺ) placed kindness to parents second only to prayer. The Qur'an puts kindness to parents in the same sentence as the command to worship God (﷾) alone. It forbids even a word of contempt to a parent in old age. It tells the child to be humble with them and to pray for God's (﷾) mercy on them. Asked who most deserves a person's care, the Prophet (ﷺ) said "your mother" three times, and then "your father". So caring for a frail parent is seen as an honour and a way to Paradise, not a burden. In much of the Muslim world, the old are cared for at home by their children.
Full explanation
What the Qur'an commands
The Qur'an joins duty to parents to the worship of God (﷾). It names old age, when a parent may become difficult, and forbids even the smallest sound of impatience. It reminds the child of the mother's pain in pregnancy and birth, and teaches a prayer for both parents.
“Thy Lord hath decreed that ye worship none but Him, and that ye be kind to parents. Whether one or both of them attain old age in thy life, say not to them a word of contempt, nor repel them, but address them in terms of honour. And, out of kindness, lower to them the wing of humility, and say: "My Lord! bestow on them thy Mercy even as they cherished me in childhood."”
“And We have enjoined on man (to be good) to his parents: in travail upon travail did his mother bear him, and in years twain was his weaning: (hear the command), "Show gratitude to Me and to thy parents: to Me is (thy final) Goal.”
“We have enjoined on man kindness to his parents: In pain did his mother bear him, and in pain did she give him birth. The carrying of the (child) to his weaning is (a period of) thirty months. At length, when he reaches the age of full strength and attains forty years, he says, "O my Lord! Grant me that I may be grateful for Thy favour which Thou has bestowed upon me, and upon both my parents, and that I may work righteousness such as Thou mayest approve; and be gracious to me in my issue. Truly have I turned to Thee and truly do I bow (to Thee) in Islam."”
What the Prophet (ﷺ) taught
The Prophet (ﷺ) ranked goodness to parents just after prayer among the deeds God (﷾) loves. He ranked being undutiful to them among the greatest sins, next to setting up partners with God (﷾). Asked who most deserves a person's good company, he named the mother three times before naming the father. He said the person who has aged parents and still fails to earn Paradise through them is the loser.
Abdullah ibn Mas'ud asked the Prophet (ﷺ) which deed Allah loves most. He said: the prayer offered at its proper time. Asked what comes next, he said: being good and dutiful to one's parents.
Abu Bakra reports that the Prophet (ﷺ) named the greatest of the great sins: to set up partners with Allah, and to be undutiful to one's parents. He then warned against false witness.
Abu Huraira reports that a man asked the Prophet (ﷺ) who most deserves his good company. The Prophet said: your mother. The man asked three more times who came next, and the answer was your mother, then your mother, and then your father.
Abu Huraira reports that the Prophet (ﷺ) said three times: may he be humbled into the dust. Asked who, he said: the one whose parents, one or both, reach old age while he is alive, and who still does not enter Paradise.
The same command in the Torah
The command to honour father and mother stands in the Ten Commandments. This is no surprise: the same God (﷾) sent Moses and Muhammad (ﷺ). The Qur'an confirms the old command and spells out what it means when parents grow old.
“Honour thy father and thy mother: that thy days may be long upon the land which the Lord thy God giveth thee.”
Jesus the dutiful son
The Qur'an honours Jesus with this same quality. Speaking as a baby in defence of his mother, he says God (﷾) made him kind to her. The Qur'an says God (﷾) chose and purified Mary above the women of all nations. The Qur'an's Jesus is the dutiful son. A Muslim who cares for his mother in her old age follows the Jesus of the Qur'an.
“"(He) hath made me kind to my mother, and not overbearing or miserable;”
“Behold! the angels said: "O Mary! Allah hath chosen thee and purified thee- chosen thee above the women of all nations.”
Perspectives
The Islamic perspective
Serving one's parents, above all in their old age, is a command of God (﷾) and a right they hold over their children. The Prophet (ﷺ) placed it just after prayer, and the mother is honoured most.
The Qur'an commands kindness to parents alongside the worship of God (﷾) alone. It forbids even a word of contempt to them in old age.
The Qur'an points to the mother's pain in carrying and bearing the child. It gives this as a reason for gratitude to both parents.
The Prophet (ﷺ) named being good to parents as the deed Allah (﷾) loves most after prayer on time.
The Prophet (ﷺ) counted being undutiful to parents among the greatest sins, next to setting up partners with Allah (﷾).
The Prophet (ﷺ) said the mother most deserves a person's good company, naming her three times before the father.
Caring for parents in their old age is a way to Paradise, and to miss that chance is a loss.
The Qur'an presents Jesus as a son made kind to his mother. So duty to parents is part of the prophets' way.
The command to honour father and mother also stands in the Ten Commandments.