Here is an answer which is only indicative but not the most definite one
According to me,the parents who don't have a son, but have a daughter, have the following two options:
1)Adopt a son.
Manu Smriti 9.141. Of the man who has an adopted (Datrima) son
possessing all good qualities, that same (son) shall take the
inheritance, though brought from another family.
Since,the adopted son is inheriting its his duty to look after the parents.
2)Make the daughter an "appointed daughter":
Manu Smriti 9.127. He who has no son may make his daughter in the
following manner an appointed daughter (putrika, saying to her
husband), ’The (male) child, born of her, shall perform my funeral
Now,an appointed daughter's son is as good as a son's son.
Manu 9.133. Between a son’s son and the son of an (appointed) daughter
there is no difference, neither with respect to worldly matters nor to
sacred duties; for their father and mother both sprang from the body
of the same (man)
So,i guess its the duty of the the appointed daughter to look after her parents like a son and thereafter the responsibility shifts on her son.
Now,in case, she(the appointed daughter) dies without a male issue ,then the duty shifts on her husband.The following verse indicate only that:
Manu 9.135. But if an appointed daughter by accident dies without
(leaving) a son, the husband of the appointed daughter may, without
hesitation, take that estate.
And,in the case,when the old parents don't have anyone , neither a daughter nor an adopted son ,then the duty(of looking after them) rests on their relatives etc.
The father, the mother, the preceptor, the wife, the children, the
poor people, the dependants, the incomers and the guests, are spoken
of as the Poshyas (i.e., those who should be supported .
(29)Kinsmen,relatives, those suffering from diseases, who have none to look after them, those who seek refuge, and others having no means, are also
spoken of as the Poshyas
(30) To support the Poshyas, is the most
excellent expedient for attaining to the celestial region(Daksha Smriti)