Commodification and Objectification of the Working Class
I find the commodification of family acts of care an interesting and controversial component of the Capitalist society. As you point out, parents are now paying others to do the labour of caring for their children, so that they can themselves engage in the labour market (outside of the home). Historically, it was common for the mother of the family to stay home and take on the labour of caring for the family and the home.
Although this is in many ways seen as an "ideal" as it allows for the mother and children to ultimately bond, the current emphasis on day care or nannies for such duties is largely because more women are in the labour market and have jobs outside of the home. This is in some ways beneficial, as it allows for the family to have a greater wealth of resources, but as well, it dislocates the mother from the child(ren) to an extent and also may create the "double shift" of work in the labour market as well as work in the home for many mothers.
In this way, it is interesting to look into how family has been commodified over the last century or so, and also aligning that with the introduction of more women into the paid labour market since WW2, which ties into Marx's concepts of commodification and objectification in the working class as you point out. However, I think this could be applied to a middle class or even upper- middle class in a contemporary context as the double-shift and paid childcare situation is not exclusive to the working class.