### Apache Airflow version

2.6.3

### What happened

When running multiple standalone DAG processors with separate DAG directories using the `--subdir` argument the processors keep deactivating each other's DAGs (and reactivating their own).

After stepping through the code with a debugger I think the issue is that the calls [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L794) and [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L798) have no awareness of the DAG directories.

### What you think should happen instead

The DAG processors should not touch each other's DAGs in the metadata DB.

### How to reproduce

Start two or more standalone DAG processors with separate DAG directories and observe (e.g. via the UI) how the list of active DAGs keeps changing constantly.

### Operating System

Linux 94b223524983 6.1.32-0-virt #1-Alpine SMP PREEMPT_DYNAMIC Mon, 05 Jun 2023 09:39:09 +0000 x86_64 x86_64 x86_64 GNU/Linux

### Versions of Apache Airflow Providers

_No response_

### Deployment

Other Docker-based deployment

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

"""
        Ensure that DAGs are marked inactive when the file is parsed but the
        DagModel.last_parsed_time is not updated.
        """
        manager = DagProcessorJobRunner(
            job=Job(),
            processor=DagFileProcessorManager(
                dag_directory="directory",
                max_runs=1,
                processor_timeout=timedelta(minutes=10),
                signal_conn=MagicMock(),
                dag_ids=[],
                pickle_dags=False,
                async_mode=True,
            ),
        )
        test_dag_path = str(TEST_DAG_FOLDER / "test_example_bash_operator.py")
        dagbag = DagBag(test_dag_path, read_dags_from_db=False, include_examples=False)
        with create_session() as session:
            
            dag = dagbag.get_dag("test_example_bash_operator")
            dag.last_parsed_time = timezone.utcnow()
            dag.sync_to_db()
            SerializedDagModel.write_dag(dag)
            
            stat = DagFileStat(
                num_dags=1,