This study questions the fMRI method itself, not the statistical analysis (you're right that the dead salmon study was challenging the way statistical analysis is done). Basically, this study claims that the association between the BOLD signal measured by fMRI and actual brain activity is quite weak, and they are even anti-correlated in 40% of cases.

There is no statistical analysis that can save you if your interpretation of a signal is wrong (for example, you can't get information about personality from phrenology, regardless of what statistical analysis you try to apply to the data). That's not to say that we need to just trust this study implicitly - I'm just trying to describe how serious of a problem to the field their claim is.