Two of the hardest problems in English-language parsing are prepositional phrase attachment and scope of conjunction. For PP attachment, the problem is to figure out how a phrase-final prepositional phrase relates to the rest of the sentence — the classic example is "I saw a man in the park with a telescope". For conjunction scope, the problem is to figure out just what phrases an instance of and is being used to combine.
The title of a recent article offers some lovely examples of the problems that these ambiguities can cause: Suresh Naidu and Noam Yuchtman, "Back to the future? Lessons on inequality, labour markets, and conflict from the Gilded Age, for the present", VOX 8/23/2016. The second phrase includes three ambiguous prepositions (on, from, and for) and one conjunction (and), and has more syntactically-valid interpretations than you're likely to be able to imagine unless you're familiar with the problems of automatic parsing.