Part 1 and 2 define the codestream and file format, respectively. They are both finalized at the technical level (the ISO process is still ongoing, but there is no more opportunity for technical changes, the JPEG committee has approved the final draft). So it is ready for use now: the bitstream has been frozen since January, free and open source reference software is available.

Part 3 will describe conformance testing (how to verify that an alternative decoder implementation is in fact correct), and part 4 will just be just a snapshot of the reference software that gets archived by ISO, but for all practical purposes you should just get the most recent git version. Parts 3 and 4 are not at all needed to start using JPEG XL.