That wasn't the thrust of the English language piece. I admit to getting only 10-15% of the Swedish version, so I can't comment there.

As to equal levels/values:

It is a hallmark of laboratory research that, in order to make a claim of causality, the research design must eliminate other possible sources that explain the observed differences in the dependent variable under examination so as to isolate the independent variable as the source of said differences. This ideal is rarely perfectly achieved, but does stand as the model for research. Correlational data can point out independent variables that MAY prove to have causal implications, but of itself can only assert that certain variables correlate with one another and that there is some relationship between them. That relationship may or may not be directly causal.

If the sexuality of the parents is irrelevant to the psychological health and development of a child, that would tend to support your position. Prima facie, that would seem to be the case given these results. Certainly in issue worthy of further study.

"religious or arcane" moral beliefs?

Did you mean "archaic" as in out-dated or old-fashioned? "Arcane" refers to something that is magical, mysterious, or requires some kind of secret knowledge. Public morality can't be any of those.