What are applications of the levels of the Chomsky hierarchy?
First, let's be clear, there are two separate types of languages, Turing Decidable (also called recursive), and Turing Recognizable (also called semi-decidable or recursively-enumerable).

The "application" of decidable languages is all of computer science. It's a huge class that includes exactly the problems that can be solved by algorithms. So you're not going to find specific applications like you would for compilers and CFGs, because the applications covered by decidable languages are "everything else." If you can write code for it, it's in this class.

Recursively-enumerable problems are ones where you know a solution when you see one, but might never know when you're out of potential solutions. You always halt on a "yes" answer, but might never halt for "no". This class is basically first-order theorem proving: we recognize proofs when we see them, but with quantifiers, there's no way to know if no proof exists.

It's also worth mentioning that parsers are CFG recognizers. Other parts of compilers, like translation and type-checking, are definitely not context-free.