You are correct - retirees together with the poor (90% of whom don't work full time).
It would not be necessary for young people to fund their parent's retirement, their parents would not necessarily need to retire. A large number of older people are capable of working, and the latest projections suggest that this trend will only increase in the future:
Spending on SSDI and all forms of welfare is dwarfed by retiree payments.
We likely agree on ratcheting up the retirement age.
This argument plays out like a chess endgame. Can we cut to the chase and acknowledge the tradeoff that's being made? At some level, we as a society are trading units of Mankiw happiness for units of peace of mind about retirement.
We certainly do it inefficiently, and, in particular, we drag alone a whole class of free riders who create a drag on the operation of our local and state governments. But even if we optimized all that away, the same fundamental response would apply to Mankiw's argument.
I am not simply pointing out that "taxes buy stuff" --- though that is also a valid point to make. I am saying that Mankiw ignores an offsetting form of incentive to work and (particularly) take risks: the well-being of our close relatives.
It would not be necessary for young people to fund their parent's retirement, their parents would not necessarily need to retire. A large number of older people are capable of working, and the latest projections suggest that this trend will only increase in the future:
http://www.sciencemag.org/cgi/content/summary/329/5997/1287