What first comes to mind is using the ShellExecute API call. Unfortunately, Control Panel Applets are not applications and cannot be opened using the ShellExecute call in a similiar way we would use it to start Windows Explorer, for example.

Control_RunDLL
To be able to execute Applets, a special purpose function named 'Control_RunDLL' has to be imported from the Shell32.DLL library (part of the Windows API).
This is what the importing looks like:

procedure Control_RunDLL(
      hwnd: THandle;  
      hInst: THandle; 
      CmdLine: PChar; 
      CmdShow: integer
      ); stdcall; external 'Shell32.dll';

The CmdLine will look different for each individual applet. Also, there is a tab number option that can be used with some applets to show a particular tab on the Applet.

Here are some examples:

//call the "Game Controllers" Applet
Control_RunDLL(Application.Handle, 0, PChar('joy.cpl'), SW_SHOW)

//call the "Keyboard Properties" Applet (contained inside the Main.cpl file)
Control_RunDLL(Application.Handle, 0, PChar('Main.cpl,Keyboard'), SW_SHOW)

//call the "Mouse Properties" Applet (contained inside the Main.cpl file)
Control_RunDLL(Application.Handle, 0, PChar('Main.cpl,Mouse'), SW_SHOW)

//call the Screen Saver tab of the "Display Properties" Applet
Control_RunDLL(Application.Handle, 0, PChar('Desk.cpl,,1'), SW_SHOW)

Here's how the call to Control_RunDLL looks inside our Delphi clone of the Control Panel folder:

procedure TMainForm.btnExecuteClick(Sender: TObject);
var
  li : TListItem;
begin
  li:=lvCPL.Selected;
  if li <> nil then
  begin
    Control_RunDLL(Application.Handle, 0, 
                   PChar(li.SubItems[1] + ',' + li.Caption), 
                   SW_SHOW);
  end;
end;

Note: the above procedure handles the OnClick event of the btnExecute TButton we added to our form. Also, the above code cannot be used to show a particular tab on the Applet.

ShellExecute?
I've said that you cannot use the ShellExecute API to execute an Applet. Well, you can. First we build a 'RunControlPanelApplet' function

function RunControlPanelApplet(const sAppletFileName : string) : integer;
begin
 ShellExecute
    (Handle, PChar('open'), 
    PChar('rundll32.exe'), 
    PChar('shell32.dll,Control_RunDLL ' + PChar(sAppletFileName)), 
    nil, SW_NORMAL);
end;

Then call it like:

RunControlPanelApplet('Main.cpl,Mouse');

And here's the Control Panel Applet, called from Delphi code:

That's it!
All set. You now have a working version of the Control Panel folder. If you have any questions, do not hesitate to post on the Delphi Programming Forum.

Of course, don't forget to download all the source code for this project...