### 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)

deactivated = (
                session.query(DagModel)
                .filter(DagModel.dag_id.in_(to_deactivate))
                .update({DagModel.is_active: False}, synchronize_session="fetch")
            )
            if deactivated:
                cls.logger().info("Deactivated %i DAGs which are no longer present in file.", deactivated)
            for dag_id in to_deactivate:
                SerializedDagModel.remove_dag(dag_id)
                cls.logger().info("Deleted DAG %s in serialized_dag table", dag_id)
    def _run_parsing_loop(self):
        
        if self._async_mode:
            poll_time = 0.0
        else:
            poll_time = None
        self._refresh_dag_dir()
        self.prepare_file_path_queue()
        max_callbacks_per_loop = conf.getint("scheduler", "max_callbacks_per_loop")
        if self._async_mode:
            
            
            self.start_new_processes()
        while True:
            loop_start_time = time.monotonic()
            ready = multiprocessing.connection.wait(self.waitables.keys(), timeout=poll_time)
            self.heartbeat()
            if self._direct_scheduler_conn is not None and self._direct_scheduler_conn in ready:
                agent_signal = self._direct_scheduler_conn.recv()
                self.log.debug("Received %s signal from DagFileProcessorAgent", agent_signal)