If you only want to run parser tests from a different file, there's no need to create a new class.
You simply add in the main file: $wgParserTestFiles[] = dirname( __FILE__ ) . "/lstParserTests.txt"; (already in lst.php)
It will be automagically picked when running the phpunit tests (if you have the extension enabled in LocalSettings, which would be a precondition for them to work).
You can also run them with tests/parserTests.php
With phpunit: $ make destructive
Tests: 5125, Assertions: 927178, Failures: 2, Incomplete: 3, Skipped: 5. Tests: 5125, Assertions: 880313, Failures: 2, Incomplete: 3, Skipped: 5.
Add lst to LocalSettings.
$ make destructive Tests: 5157, Assertions: 883148, Failures: 2, Incomplete: 3, Skipped: 5. Tests: 5157, Assertions: 942272, Failures: 2, Incomplete: 3, Skipped: 5. Tests: 5157, Assertions: 913264, Failures: 2, Incomplete: 3, Skipped: 5.
You can also run make parser if you prefer to run less tests. (no, I don't know why would the number of assertions randomly change between runs with the same config...)
And fyi, they do pass.