Extend the component that you want to draw on, call
super.paintComponent() from the new components paintComponent method and
then draw away.  This isn't going to work if it is some sort of active
component like a button or something.

import java.awt.*;
import javax.swing.*;

public class test {
    public static void main(String[] args) {
        Runnable r = new Runnable() {
            public void run() {
                class OriginalThing extends JPanel {
                    public OriginalThing() {
                        setPreferredSize(new Dimension(400,300));
                    }
                    public void paintComponent(Graphics g) {
                        g.setColor(Color.BLUE);
                        g.fillRect(0,0,getWidth(),getHeight());
                    }
                }
                class ThingIWantToDrawOnToo extends OriginalThing {
                    public void paintComponent(Graphics g) {
                        super.paintComponent(g);
                        g.setColor(Color.RED);
                        g.drawString("I drew this on the original!",20,30);
                    }
                }

                JFrame f = new JFrame();
                f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);

                f.add(new ThingIWantToDrawOnToo());

                f.pack();
                f.setVisible(true);
            }
        };
        EventQueue.invokeLater(r);
    }
}

Think of the Graphics2D as a pencil or paintbrush that draws on a
particular drawing surface, not as the drawing surface itself.

Sounds to me like what you might need is to get a BufferedImage of the
appropriate size and type, get a Graphics object from it, and with that
Graphics object call your panel's paintComponent. That will give you a
copy of the panel's rendering in the BufferedImage so you can work with it.