As a career mathematician / software developer, NOT prone to conspiracy theories, this study nevertheless got my statistical Spidey sense tingling. If I were determined to rig an election, particularly through electronic voting/tallying, this is EXACTLY how I would do it; selectively target larger precincts, because the vote flipping is less likely to be noticed there. (And more importantly, because spot-tests of the system are unlikely to cast enough votes to trigger the mechanism.)
That said, the study is sloppily done, not peer-reviewed, and prone to accusations of cherry-picking. They claim to have replicated their results all across the country, but provide no data to back this up. (E.g. they should show a scatterplot showing voting mechanism vs. "anomaly" strength, for a large number of states or counties.) And their shining example, the 2012 Iowa Primaries (actually Caucuses), DID use paper ballots and precinct-level tallying, yet still showed the anomaly. I'd like to hear their explanation for how they think the fraud could have crept in here. They also use Duval County, FL 2012 Primaries as another example of the anomaly, but paper ballots were used there as well. I don't know if the tallying was per-precinct or centralized for that election; if it were centralized, the fraud could easily happen there because it's a single point of failure.
More than anything, I would LOVE to get Nate Silver's take on this study. Perhaps he would have some intuition for how the precinct size / vote correlation might have arisen "naturally," and presumably he has access to the databases required to re-run the study on a larger scale. Either way, it's absolutely clear that paper ballots and transparent precinct-level tallying are essential to ensure fair elections. They can pry my cold, dead trees from my cold, dead hands! ;-)