To be more specific (and your parser example actually makes it very clear), exceptions are a form of an error monad. The benefit that you describe - the ability to automatically flow throw error values without having to check for them at every boundary - is exactly what a monad provides.
The problem with exceptions is that they're not type-checked in most languages - you can throw whatever you want, and the type system doesn't reflect that, so there's no way to statically determine that all expected exceptions are properly handled.
They're (partially) type-checked in Java, but it's extremely annoying, because there's no ability to express that part of the type in a way that lets you write methods that are generic with respect to exceptions thrown by other code (e.g. there's no way to describe a method `f(T x)` such that it throws `E` plus everything that `x.g()` may throw, for any `T`).
The problem with exceptions is that they're not type-checked in most languages - you can throw whatever you want, and the type system doesn't reflect that, so there's no way to statically determine that all expected exceptions are properly handled.
They're (partially) type-checked in Java, but it's extremely annoying, because there's no ability to express that part of the type in a way that lets you write methods that are generic with respect to exceptions thrown by other code (e.g. there's no way to describe a method `f(T x)` such that it throws `E` plus everything that `x.g()` may throw, for any `T`).