Any unit of measurement would have been arbitrary, and twelfths at least lets you refer to the first third, quarter or half. No idea whether that was the rationale, but I don't think any other division would have seriously affected the point the article is making.
Three act structure with each act divided into quarters.
Though, to be honest, three acts is kinda misleading since most second acts have a big turning point in the middle. So while yeah, there's a beginning, middle, and end, that middle is really two acts.
12 parts be quarters split into three as easily as thirds split into four.
Though in any case it would be nice for the article to state why that division was used (even if the reason is completely arbitrary - at least then we'd know to stop guessing!).