Did you reverse this? e.g. compression is useful for POS tagging, etc.

No, the original is correct. Let's say you want to compress a text corpus. Then you need a good probability model P(s), of the probability of a sentence, that assigns high probability to the sentences in the corpus. Since most sentences follow grammatical rules, the model P(s) will need to reflect those rules in some way. The better you understand the rules, the better your model will be, and the shorter the resulting codelength. So because parsing (and POS tagging, etc) is useful for compression, the compression principle allows you to evaluate your understanding of parsing (etc).