So then the parser ["read"] just has to be able to take a textual representation of s-exprs, and turn it into their in-memory representation (and run any reader macros).
But anyway, I think people say that because the job of the parser (especially if you ignore reader macros, and if you just look at it on a surface level) seems very simplified compared to what the parser for a lot of languages does to generate the AST, because you already have a textual representation of the AST.
On the other hand, lisp is often described as having read-time, compile-time, and run-time, and compile-time itself (sometimes) gets described as having a macro-expansion phase, and then a later phase, and handling a lot of what is syntax in other languages (I think) gets put off until macro-expansion time.