I think your source's claim that "The planet will never see more babies than it has in 2013" is open to question.
To just use the US as an example: mainstream Americans have a declining fertility rate, and we have no reason to think that is going to reverse in the foreseeable future. Immigrants have higher fertility, but they tend to converge to the mainstream after a generation or two. So, that seems to support the article's contention, at least for the US – and if we look at other countries, we observe things are broadly similar, so that in turn supports that contention for the planet as a whole.
However, we also observe in the US small, ultra-conservative religious minorities, such as the Amish and ultra-Orthodox Jews, who still have a high fertility rate, and also have high youth retention (80-90% of their children will stay in the group as adults). Now, even though these groups are a tiny percentage of the population, the miracle of exponential growth means that within 2-3 centuries, they could be the majority of the US population, which could then result in a national baby boom. And, eventually, as they spread across the globe (they exist to varying degrees in other countries too), a global baby boom. So 2013's "peak baby" may end up being surpassed.
Of course, there is no guarantee that is going to happen – maybe they'll get to a certain size, and then they'll stop growing. And of course, it is physically impossible to sustain exponential growth indefinitely. However, it remains a possible future that they won't stop growing until after they've gotten so big that 2013 turns out not to be the year of "peak baby" after all. Maybe it will actually be 2213 or 2313 instead. Our descendants (if one happens to have them) will find out.