$y^{2}=4 x \,\,\,\,\,\dots(i)$
$x^{2}=-32 y \,\,\,\,\, \ldots(2)$
$m$ be slope of common tangent
Equation of tangent $(1)$
$y=m x+\frac{1}{m} \,\,\,\,\dots(i)$
Equation of tangent (2)
$y=m x+8 m^{2} \,\,\,\,\dots(iii)$
(i) and (ii) are identical
$\frac{1}{m}=8 m^{2} $
$\Rightarrow m^{3}=\frac{1}{8}$
$ m=\frac{1}{2}$ Alternative method:
Let tangent to $y^{2}=4x$ be
$y=m x+\frac{1}{m}$
as this is also tangent to $x^{2}=-32y$
Solving $x^{2}+32 m x+\frac{32}{m}=0$
Since roots are equal
$\therefore \, D=0$
$\Rightarrow (32)^{2}-4 \times \frac{32}{m}=0 $
$\Rightarrow m^{3}=\frac{4}{32} $
$\Rightarrow m=\frac{1}{2}$