Correct solutions were received from Chen from The Chinese High School Singapore, Andrei of School 205 Bucharest, Mary of Brirchwood High School (whose solutions is used as the basis of the one below) and Sana, Jenny, Chris and Rosion of Madras College, St. Andrews.
l is a 1/3 the length of the rectangle
w is a1/3 the width of the rectangle
area of the rectangle = 3l x 3w = 9lw
Area triangle A = Area of triangle C = lw
Area triangle B = Area of triangle D= lw
Therefore area of the parallelogram = 9lw - 4lw = 5lw
Therefore ratio of the area of the parallelogram to the area of the rectangle is 5lw : 9lw = 5:9