You are not using the module as it is intended. It's an extension to the SQL::Parser module that is used by DBDs to implement SQL syntax and functions. The documentation states:
This module contains the built-in functions for SQL::Parser and SQL::Statement. All of the functions are also available in any DBDs that subclass those modules (e.g. DBD::CSV, DBD::DBM, DBD::File, DBD::AnyData, DBD::Excel, etc.).
To me, that sounds like it has nothing to do with mysql. The DBD::mysql is used as a wrapper around a real mysql database. It just connects, forwards the queries, prepares them and so on. It does not parse them, however. That's what the database is doing, just like if you were to connect to an Oracle or MSSQL database. In those cases, the driver is like an API wrapper.
On the other hand, DBD::CSV and the others mentioned above are parsing the sql themselves. DBD::CSV opens a CSV file, understands the SQL and (very simply described) translates it into Perl code to look for stuff in the CSV data. The same goes for the others. They use SQL::Parser to make sense of the SQL. SQL::Statement::Functions seems to be the module that provides the equivalent Perl functions for the SQL.
When using SQL::Statement/SQL::Parser directly to parse SQL ...