How to call parent activity function from ASyncTask?

Use Interface

Follow these steps:

1) Create an interface

public interface AsyncTaskListener{

   public void updateResult(String result);

}

2) Use the listener in your AsyncTask

DownloadSongTask extends AsyncTask<String,Integer,String>{

   private AsyncTaskListener listener;

   public DownloadSongTask(Context context)
   {
       listener= (AsyncTaskListener)context;    // Typecast 
   }

   @Override
   public void doInbackGround(String... params)
   {
       // Download code
       int downloadPerc = // calculate that
       publish(downloadPerc);   
   }

   @Override
   public void onPostExecute(String result)
   {
       listener.updateResult(String result);  // Use it 
   }

}

3) Implement the interface in your Activity and Override the interface method

  public class YourActivity extends AppcompatActivity implements AsyncTaskListener{

  // Activity code //
  new  DownloadSongTask(this).execute("Paradise.mp3");  // this is how you start Task

  public void yourMethod(String arg)
  {
    // Your method related Stuff
  }

  @Override 
  public void updateResult(String result){
        yourMethod(result);
  }

}

Advantege of using interface?

This seems a lengthy approach at first but if you use this approach

You can make a loosely coupled AsyncTask. Which means you can use same AsyncTask with any Activity in Future without even changing code in your AsyncTask.

Relevant Links:

For better understanding you can read this ANSWER

When you create the AsyncTask, you can add a new constructor to it, and pass in a reference to the Activity:

AsyncTask myTask = new MyTask(this);

And then from the onPostExecute() method in the AsyncTask you can call the method on the Activity.

public class MyTask extends AsyncTask<String, String, String>
{
    public MyActivity activity;

    public MyTask(MyActivity a)
    {
        this.activity = a;
    }

    //  ......

    protected void onPostExecute(String result)
    {
        activity.myMethod();
    }
}