From external point (x1,y1), the chord of contact to $y^2$=4ax is T=0: yy1=2a(x+x1). The pair of tangents is $T^2$=S*S1. The chord with midpoint (x1,y1) satisfies T=S1. These are the same T=0 substitution rules as for circles, applied to the parabola equation.