We can use basic coordinate geometry to find the equation of the normals. Let $y_p$, $y_q$ and $y_r$ be the y coordinates of the points p, q and r on the parabola. Then $\frac{y_p^2}{4a}$, $\frac{y_q^2}{4a}$ and $\frac{y_r^2}{4a}$ are the x coordinates. Then the equation of the normal at p is $$y = -\frac{y_px}{2a} +\frac{y_p^3}{8a^2} + y_p$$ and the normal at q is $$y = -\frac{y_qx}{2a} +\frac{y_q^3}{8a^2} + y_q$$
Solving these two equations we have $$x=\frac{y_q^2+y_py_q+y_p^2}{4a}+2a$$
This equals k per question.
Next we use the coordinates of points p and q to find the equation of $pq$. We have $$x=\frac{(y_p+y_q)y - y_py_q}{4a}$$. Then put this x in the given parabola we have $$y^2 - 4(y_p+y_q)y + 4y_py_q - 16a(2a-k) =0$$
The last step is to prove this quadratic equation has only one root in order to be a tangent. Since we have $$16(y_p+y_q)^2-16y_py_q+64a(2a-k)=0$$ because $$y_q^2+y_py_q+y_p^2=4a(k-2a)$$ the line pq cuts the given parabola at only one point and is a tangent to the second given parabola.
This is also true for the the line joining qr and $pr$.