Earlier quoted context omitted.
Another theory: As this article points out a lot of woman aren't marrying or having kids because "being both employed and married is tough in Asia.". In other words, these countries have enough equality where women are allowed in the labor force but not enough where the society has set up strong institutions to deal with women in the labor force. Another economist article dealt with the impossibility of finding day c…
I'm not sure how your policy recommendation relates to your initial observation (gender equality -> fertility). How does forcing the childless (note: people of both genders can be childless) to subsidize parents promote gender equality? (I'm not disagreeing with your suggesting that you can subsidize something to get more of it, I'm just trying to figure out how your last paragraph relates to the rest of your post.)
First note that the childless already massively subsidize parents through taxes that pay for education. Also note that I am considering replacement levels of fertility desirable.
The cost of being a mother, biologically, is vastly higher than being a father. This is further enhanced by cultural expectations for the mother to do most of the child raising.
In other words, female parents take a far greater hit than male ones. (Indeed, child raising explains almost all of the male-female wage gap). Subsidizing parents effectively subsidizes women far more; it's basically an affirmative action program to compensate a disadvantage dealt by biology and society. (How the subsidy flows is tricky. As most people end up having kids, this subsidy is effectively mostly a transfer of wealth from fathers to mothers, across generations).
Some policies that may be clearer are government promotion of fathers taking care of kids via both advertisement and paternity leave. This breaks down traditional gender roles (promotes equality), while reducing the burden on the mother.