### Body

Same as https://github.com/apache/airflow/issues/33876 but for `publish-docs` command.

### Committer

- [X] I acknowledge that I am a maintainer/committer of the Apache Airflow project.
package_list: list[str],
    airflow_site_directory: str,
    override_versioned: bool,
    include_success_outputs: bool,
    parallelism: int,
    skip_cleanup: bool,
    debug_resources: bool,
):
    """Run docs publishing in parallel"""
    with ci_group("Publishing docs for packages"):
        all_params = [f"Publishing docs {package_name}" for package_name in package_list]
        with run_with_pool(
            parallelism=parallelism,
            all_params=all_params,
            debug_resources=debug_resources,
            progress_matcher=GenericRegexpProgressMatcher(
                regexp=PUBLISHING_DOCS_PROGRESS_MATCHER, lines_to_search=6
            ),
        ) as (pool, outputs):
            results = [
                pool.apply_async(
                    run_docs_publishing,
                    kwds={
                        "package_name": package_name,
                        "airflow_site_directory": airflow_site_directory,
                        "override_versioned": override_versioned,
                        "output": outputs[index],
                        "verbose": get_verbose(),
                    },
                )