I may have made that jump too quick.

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.

>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.

This isn't as simple as that. The tax revenue used for education in Texas comes from property taxes. The majority of that is from homeowners. The overwhelming majority of homeowners have or had children in the home at some point. Aside from those facts, would you like to live in a society where the children were not educated? I think you would find that the reduction in crime and the boost to the cheap labor force would be worth the investment.

The most important point this should make is: all of these things are fabulously complicated and interrelated.

Ok, so by "gender equality", you merely mean "statistically similar representation of women in assorted reference classes" rather than "equal treatment". (The latter is what I originally thought you meant.)

Of course, I'd be rather hard pressed to see what that has to do with fertility. I'd suggest maybe that there is a simpler explanation for the phenomenon you observed - the countries with gender equality also seem to have lots of subsidies for parenthood. Most likely the subsidies are the cause of fertility.

If the subsidies are the actual cause, we could probably get higher fertility by subsidizing parenthood together with higher societal expectations of maternal activities from women. The carrot of free money for the fertile life path, and the stick of lower social status for the less fertile path.

(Not that I advocate this course of action, but I'm not in favor of encouraging fertility in any case.)