CAPSLOCKENGAGED

Here is an application of parse trees: sentiment analysis with recursive neural networks based on how components of the parse tree combine to create the overall meaning.

http://nlp.stanford.edu/~socherr/EMNLP2013_RNTN.pdf

They are useful as a preprocessing step for a lot of downstream NLP tasks. It shouldn't be hard to find more papers that take advantage of the tree structure of language.

That is not necessarily true. The problem of ambiguity is fundamental to natural language processing, and a lot of research goes in to addressing it. If we also see a sentence where unambiguously the word "cat" is the subject of the verb "meow", then this could give our parser clues about ambiguous attachment or, in this case, anaphora resolution (to what does "it" refer). In any intro NLP class, you will learn about lexicalized parsing, which takes the head word of the phrase into account when making parsing decisions. I haven't read the paper on this parser yet, but I don't think it is hard to see that your sentence could be accurately parsed given enough data. Look up "word embeddings" for instance, which are fundamental to deep learning for NLP and could probably be trained to assist in disambiguating anaphora or attachment.