What the writings by Sullins say
Invisible Victims: Delayed Onset Depression among Adults with Same-Sex Parents is available online:
Retrospective questions at Waves III and IV asked about adult mistreatment during childhood, including whether a parent or caregiver had “slapped, hit or kicked you,” said “things that hurt your feelings or made you feel you were not wanted or loved,” or “touched you in a sexual way, forced you to touch him or her in a sexual way, or forced you to have sex relations.” Respondents reporting any physical, verbal, or sexual abuse at either Wave were coded positive for abuse victimization. Four-fifths (79%, 95% CI 77–80) of reported mistreatment was verbal abuse.
Ninety percent of the same-sex parented children reported parental abuse at Wave III
The 92% figure is from this table.
The percentage for opposite-sex parents was 58.2%.
Implication of the poster
The image in the poster implies physical abuse by men.
However, Sullins only looked at 20 cases, 17 of which were children of lesbians. Additionally, 79% of the abuse was verbal abuse.
Criticism of Sullins and his writing
The same journal that published his writings, published a comment on those which criticizes his methodology:
Sullins achieves this through a crucial elision between households in which a child spent some time in a home headed by a same-sex couple and families in which a child was actually raised, from birth, by a stable same-sex couple, a situation more auspicious for healthy child development. This conflation of household stability with parent gender fatally mars his conclusions, which are much more damning of gay and lesbian parenting than are warranted by his data.
Sullins claims that his study examines “children raised by same-sex parents into early adulthood.” But in fact, he has zero basis to draw this conclusion, as he is applying a wholly untenable definition of “raised by.” All he knows about his dataset is that his subjects, who ranged in age from 12 to 18, spent some of their teenage years with a parent who at some point had a same-sex partner. Since we do not know if that partner was ever actually a parent, legally or otherwise, it is inaccurate to characterize such households as “same-sex parented” as Sullins does eleven times.
Not only is there no basis to conclude that these subjects were raised by same-sex parents, but also there is every reason to believe they likely were not.
This descriptor, of course, is the key variable in the discourse on optimal child-rearing because of the well-established fact that children who experience divorce or other family disruptions are at higher risk for a number of disadvantages, including the ones that Sullins inaccurately associates with “same-sex parented” households.
The main point here is that Sullins uses a poor definition of "raised by" which doesn't actually mean "raised by", and which includes a disproportionate number of children affected by divorce for same-sex couples.
As children were not asked who abused them, in can happen that children that Sullins describes as "raised by" same sex couples were abused by their other parent.
As T. Sar pointed out in the comments, further criticism - including criticism of the journal as low quality and pay to publish - is formulated in this Slate article.
The publisher also commented on the publication:
In June 2016, several readers raised concerns about this article. At that time, we evaluated the article’s peer review process and brought several concerns to the handling editor’s attention. These included: the study’s small sample of same-sex parents, the lack of discussion of other influences such as family breakup on the wellbeing of the children included in the study, the implied causation in the title “Invisible Victims,” and the potential conflict of interest implied by the author’s position as a Catholic priest.
Paul Sullins is in fact not just a priest, he works with the Family Research Council, which is designated as a hate group by the SPLC. The FRC thinks that homosexuality is "harmful", "can never be affirmed", and is "by definition unnatural, and as such is associated with negative physical and psychological health effects". They oppose acceptance of homosexuality in "law, in the media, and in schools".
The APA answers the broader question regarding same-sex parenting:
On the basis of a remarkably consistent body of research on lesbian and gay parents and their children, the American Psychological Association (APA) and other health professional and scientific organizations have concluded that there is no scientific evidence that parenting effectiveness is related to parental sexual orientation. That is, lesbian and gay parents are as likely as heterosexual parents to provide supportive and healthy environments for their children. This body of research has shown that the adjustment, development and psychological well-being of children are unrelated to parental sexual orientation and that the children of lesbian and gay parents are as likely as those of heterosexual parents to flourish.