What exactly do you find to be "bullshit" about this study? Be specific.
1. It has not been peer reviewed. PLOS states if there was a peer review the reviewer would be named.
2. Participants were not randomly chosen but were persons who chose to visit a website.
3. The sample sizes are quite different - 1174 liberals, 500 conservatives.
4. On the 7 point scale, "our sample of conservatives contains a higher proportion of self-described slight conservatives than a representative population would." The "mean" of conservatives in the study would then be moved toward less conservative than the population in general.
5. It allows each group to self identify it's morality and then compares the other group's view. If one group is more prone to lying about their personal morality, then the comparison is only what each side perceives their morality to be not what their actual positions are.
6. The accuracy is determined based on the mean of each group. We already know that the conservative mean is different than what it is in the general population. There is no reference to how the liberal sample compares to the general population.
7. There is no attempt to correct the data to represent the general population but conclusions are drawn that are attributed to the general population.
One would expect that extreme liberals and extreme conservatives would be more likely to mischaracterize the moral position of the other side. This study fails to look at the various levels of conservative/liberal on their scale and doesn't attempt to correct for bias in the answers based on not being representative of the general population. While it does raise some interesting questions, it does little to answer them. It also isn't close to what the OP claimed was in this study.