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I'm taking a look at the source paper and it looks like Sulins is just comparing children from same sex couples and children from other sex couples, without controlling for adoption.
Children who were adopted either had abusive parents, or lived without parents for some time, and are much more likely to develop those kinds of mental illnesses. That by itself could explain the correlation between same-sex parents and mental illness, because 100% of children raised by gay couples are adopted.
