This depends on the geographic region you are looking at. European families tended to be either a "nuclear family" (more in northern europe) or an "extended family" (more in southern europe). I don't know much about Southern Europe but in those extended families, which often included cousins and grandparents, children took care of their aging parents by staying together for a long time.

In the nuclear family, at least of the flemish and brabant variation, this was also an option: a parent moving back in with his children. It was however more common that the parent would have bought some kind of life insurances, commonly from the city government, which would pay the person in question a yearly sum, which would increase in amount year by year. People could of course also have saved up capital, bought properties which would bring them easy money at old age, get inheritances from recently deceased family members etc. The most common way however, especially for lower to medium classes, was the life insurances.

If you were actually dirt poor, you would have to rely on charity, which was usually quite well organised by the church in urbanised regions (again I speak of Flanders and Brabant).

I spoke of the Southern Netherlands, which was unusually highly urbanised throughout the middle ages. I am not sure to what extent this is also valid for less urbabised regions such as England.