My question is regarding the role of children towards their parents particularly in old age.
1) Firstly is there a difference in the duties towards parents for a son and a daughter? Meaning do sons and daughters have different duties towards their parents.

In South Asian culture a daughter once she marries is expected to only look after her husband's parents while no duty is expected towards her own parents.
Whereas for a son since he has to go out and earn and hence is not at home it therefore becomes his wife's duty to take care of his parents as he is financially providing for her . Is this the islamic way of life?

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All of us according to Islam have two types of duties one is obligatory and other is related to our ethics.
Parents have obligation towards their children same way for both son and daughter to care of them and protect them and provide them good education and sustainability.
According to verse 23 from chapter Isra:

وَقَضَىٰ رَبُّكَ أَلَّا تَعْبُدُوا إِلَّا إِيَّاهُ وَبِالْوَالِدَيْنِ إِحْسَانًا إِمَّا يَبْلُغَنَّ عِندَكَ الْكِبَرَ أَحَدُهُمَا أَوْ كِلَاهُمَا فَلَا تَقُل لَّهُمَا أُفٍّ وَلَا تَنْهَرْهُمَا وَقُل لَّهُمَا قَوْلًا كَرِيمًا

And your Lord has commanded that you shall not serve (any) but Him, and goodness to your parents. If either or both of them reach old age with you, say not to them (so much as) "Ugh" nor chide them, and speak to them a generous word.(17:23)

It has not mentioned and restricted for male child or female rather the verse says same duties for both male and female children to take care or their parents specially when they reached to their old age. Now it’s up to children how do they take care of their parents because taking care doesn’t meant only when they live together because ayah doesn’t mention that they have to live together or far away.

But it’s clear that it’s wajib on children without any restriction of their gender to take care of their parents. Now since our culture mostly girls leave their parental home after their marriage and most of them have no source of income so it’s practically seems hard for them to take care of their parents. But the other side mostly sons are with their parents or at their parental home it looks easy for sin to take care of his parents.
But as for as obligation of taking care of their parents concern is no doubt clear for Son.

Not if we talk about a woman and her duties towards her family when according to Islam she is not bounded to work at home rather it’s her ethical obligation to do house work while it’s husband who has duties to provide a helper for her if she is not capable but for other things like obeying her husband and fulfill his sexual needs it’s wajib on a woman to do so.
But it’s not obligatory for a woman that she has to take care of their parents ( in laws) rather her husband has to take care of his parents by any means. In short mostly it depends on our ethical obligations and for a good and healthy home environment that we have to take care of each other and do not interrupt each other unnecessarily. Also it’s better for our next generation to have a good guidance of father and grandfather moral support.

My question is regarding the role of children towards their parents particularly in old age.
1) Firstly is there a difference in the duties towards parents for a son and a daughter? Meaning do sons and daughters have different duties towards their parents.

In South Asian culture a daughter once she marries is expected to only look after her husband's parents while no duty is expected towards her own parents.
Whereas for a son since he has to go out and earn and hence is not at home it therefore becomes his wife's duty to take care of his parents as he is financially providing for her . Is this the islamic way of life?

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No there is no difference between sons and daughters when it comes to caring for their parents. It is an equally shared responsibility. The expectation that the daughter in law only serves her husband’s parents and has no duty towards her own parents is not an Islamic way of life. She has a great responsibility towards her parents. She still has a duty to serve them and help them if they need help.

Now if she lives in a culture where it will make a big problem for her if she serves her own parents, and this will create tension with her husband, then she should be wise and creative and avoid such tension. She can figure out indirect ways of helping her parents. Or serving them in a way that won’t cause tension with her in laws. But she definitely should try her best to serve her parents just like she takes care of her husband’s parents.