Child-class refering to other Child-classes

Question

Hello!
I am an inexperienced Java-programmer who have recently given the following some thought: if I have a Parent class that creates an instance of another class, and that class, in turn, needs to refer back to a sibling instance in the Parent class, what is the best way to go?
I can see possibilities with creating a static method in Parent, that returns a reference to whatever other child class you might need to access. Also, I guess you could send the Parent-class to the child class as an argument, and save it in a reference-variable to use when you need to use one of the Parents non-static methods. Probably there are more possible ways.
But what I'm asking is; which way is the most "legitime" when it comes to OOP? Surely it's butt-ugly to have a load of static methods in a Parent-class, or to propagate the Parent class through levels and levels of childs and grand-childs.
If I haven't made myself clear -- I have doubts about my English :P -- I'll try to illustrate my question with a short code-snippet below
Class Parent
Child myChild;
Sibling mySibling;
public Parent()
myChild = new Child();
mySibling = new Sibling();
// ... more follows
Class Child
public Child()
// Here, I want to somehow access "mySibling" in class Parent
// ... more follows
// ... more follows
Class Sibling
// ... more follows
}

Answer 1

As I tried to illustrate with my code snippet, the children are of separate classes. I could yet again clarify my point by giving a more concrete example, one that's the actual cause of my question.
I have a class [MainClass] that extends JFrame, implements MouseListener, and calls its own constructor in a main()-method.
The constructor then goes about creating the Frame's interior by creating instances of JPanels and, in one case, an extenden JPanel called CardPanel (implements ActionListener). So now we have three JPanels (including the extended) active:
JPanel cardPanelButtons; // is the topmost panel in the frame, and lets the user switch between the different cards in the CardPanel. It contains three buttons that are in a ButtonGroup to make sure only one of them is pressed at any one time.
CardPanel cardPanel; // is centered in the frame, and contains every interactive item in the application, except for the buttons in the cardPanelButtons-JPanel
JPanel statusPanel; // contains a non editable JTextArea for status-messages
CardPanel creates three panels in its constructor, and adds these to itself.
In one of these panels (that are selected by the user, clicking on the buttons in the cardPanelButtons-JPanel) there is a "Search"-button that when pressed, performs a Database-search and then pops up the next card in the CardPanel with the apropiate query result filled in and displayed. This is all good, however for GUI-purposes, this action requires the apropiate button in the cardPanelButtons-JPanel to be "pushed" automatically when this switch between cards take place. Since these buttons are unknown to CardPanel, it needs to somehow reach them, which is where I'm starting to wonder what the best way around this is.
Should I have the CardPanel-constructor take the parent class as an argument, and access the buttonGroup by a public getter-method in MainClass. Or should I static a public getter-method in the MainClass, letting me ignore to pass on the MainClass for reference. Should I ignore extending JPanel to create CardPanel altogether, and add everything straight into the JFrame from a MainClass-method?
What is the most correct way, or the way programmers usually go about this type of problem?
(In order to make myself perfectly clear, this reply is quite extensive. I apologize if I got a bit carried away..)