Aside from memory safety, which is solved by using a compiler that just doesn't allow unsafe memory operations (so not GCC or Clang upstream), which CVEs specifically would have been ameliorated by a parser written in Rust instead of C?
I don't see how that's possible without turning the language into something that isn't C, either by adding significant new functionality (e.g. fat pointers) or subtracting enough functionality that it's a much less capable language (e.g. disallowing dynamic memory allocation).
An important improvement over rust is that "Fil-C has no unsafe statement."
In case of fil-c, it is about 1.5-4x slower performance, and a memory overhead.
So, let's not present it as a panacea to all problems: there could good reasons to use it, but it isn't a magic trick.