A programming language implementation is either a compiler or interpreter. Title says "write your own programming language..." which implies an implementation. But that's not the case. As other commenters point out parser is a tiny, usually inconsequential part of a lang implementation.

The article contains full interpreter?

I wouldn't call it a full interpreter, it just walks expressions' AST. No way to create class/abstractions.

Nearly all mature language implementations are compilers to some extent