let the system of parallel chords have a slope m.

if (x1, y1) is the middle point of any one chord of this system then we wish to find the locus of (x1, y1).

Now equation of the chord with middle point (x1, y1) is

yy1 - 2a(x + x1) = y1^2 - 4ax1

slope of this chord = 2a/y1

but since we assumes that the slope is m we have

m = 2a/y1 or y1 = 2a/m

Hence locus of (x1, y1) is

y = 2a/m which is a line parallel to the axis of the parabola.