And getting more than 3 hours of sleep a lot is a privilege that's not available to recent parents.
I don't know the exact details of what policies are correct for paid time off, but the fact is still true that we have a strong societal incentive to handle situations like this in a way that's beneficial both to the parents and (especially) beneficial to kids -- at least if we believe that parental presence helps early development, which seems reasonably well supported.
I say this as a single person with no plans to ever marry or have children, but the reality is that being single or having kids each carries its own set of advantages and disadvantages, and I'm not sure that fairness between parents and childless individuals is necessarily the best lens to look at policy through.
There's a lot of middle ground here between status quo and what GP describes, but even in that case raising a child is still an 18+ year commitment; I don't think that a single family vacation significantly changes that.
And sure, as a non-parent I can't take a multi-month long vacation every year, but even as a recent graduate I get 25 paid vacation days per year (5 of which can be saved for a few years). So long vacations aren't a pipe dream for non-parents.
And with the way the economy currently works and the gradually increasing life expectancy, increasing the birth rate is a good thing, and generous parental benefits are a way to encourage that. I personally think it would be best if society/the economy wasn't dependant on an increasing population, but that's a separate conversation.
In the US, we have a program called WIC, to help women with infants and children with nutrition. If you are a poor man or poor single woman, this program is not available to you. I would not consider that a privilege. We are trying to help foster the next generation and improve society.