To enable parsers, you must set the value of FluentBitConfig.Spec.Service.ParsersFile to parsers.conf. Your custom parsers will be included into the built-in parser config via @INCLUDE /fluent-bit/config/parsers.conf. Note that the parsers.conf contains a few built-in parsers, for example, docker. Read parsers.conf for more information.
Check out the demo in the folder /manifests/regex-parser for how to use a custom regex parser.