I don't think perfect parity in childcare is worth some of the artificial contrivances that perfectly equitable circumstances might demand.
Modern society is not natural or normal, in a cosmic sense.
If you mean freeloading, as in skipping town and denying involvement in a pregnancy, your arguments hold water, inasmuch as putting a child up for adoption is a similar degree of freeloading.
If you mean father and mother both being equally in the child's continual presence for the first 24 months of life, I doubt humanity's general economic wealth, in aggregate, could support such practices.
Should children be raised by butlers and maids in the name of equitable career paths for parents?
Maybe we should all be raised in B.F. Skinner Boxes, and learn to walk through operant conditioning.