Thanks to Shashikant Patil for figuring out that the code needs to invalidate the MDI window.

' Make the image fit the MDI form.
Private Sub MDIForm_Resize()
Dim client_rect As RECT
Dim client_hwnd As Long

    picStretched.Move 0, 0, _
        ScaleWidth, ScaleHeight

    ' Copy the original picture into picStretched.
    picStretched.PaintPicture _
        picOriginal.Picture, _
        0, 0, _
        picStretched.ScaleWidth, _
        picStretched.ScaleHeight, _
        0, 0, _
        picOriginal.ScaleWidth, _
        picOriginal.ScaleHeight
        
    ' Set the MDI form's picture.
    Picture = picStretched.Image

    ' Invalidate the picture.
    client_hwnd = FindWindowEx(Me.hwnd, 0, "MDIClient", _
        vbNullChar)
    GetClientRect client_hwnd, client_rect
    InvalidateRect client_hwnd, client_rect, 1
End Sub