[preserves]

This is exactly the type of problem that a good parser should be able to solve, and training a parser on lots of data and throwing neural nets may indeed be a viable solution. Why wouldn't it be? The article describes how their architecture can help make sense of ambiguity.

In terms of a basic probabilistic model, P(meow | rug) would be far lower than P(meow | cat), and that alone would be enough to influence the parser to make the correct decision. Now, if the sentence were "The cat sat on the rug. It was furry", that would be more ambiguous, just like it is for an actual human to decode. But models trained on real-world data do learn about the world.