Is parenthood permissible for a utilitarian, given the greater good that could be done with the resources you'll spend on it?
'Permissibility' is a poor term to use in the context of classical act utilitarianism, as this ranks actions as better or worse and would often only consider the best action as permissible. Parenthood likely does not devote resources to their optimal target, so is not the best choice in the sense, but is not bad when compared to the many other suboptimal actions people typically take.
Some other points are covered by this article:
- The costs of parenthood are comparable to other life choices generally deemed acceptable (see source)
- Some find abstaining from parenthood so psychologically difficult that it's self-defeating
- There are significant benefits
Whether it is forbidden, required, or somewhere in between depends entirely on your cost function, especially how you view the number and quality of future lives impacting the determination of the greater good.
With certain choices (e.g. those who are here now are privileged with respect to new lives), the species may end up--if you follow it to its logical conclusion--courting extinction.
For any flavor of utilitarianism that does not risk extinction of humanity, parenthood is a good course of action for many.
(I am not aware of a systematic review of the long-term evolutionary fitness obtained by following various ethical systems. This might be nice to know, though.)