If the common tangent to the parabolas,

 Equation tangent to parabola $y^{2}=4 x$ with slope $m$ be:

$y=m x+\frac{1}{m}$ ...(i)

$\because$ Equation of tangent to $x^{2}=4 y$ with slope $m$ be :

$y=m x-a m^{2}$ .......(ii)

From eq. (i) and (ii),

$\frac{1}{m}=-m^{2} \Rightarrow m=-1$

$\therefore$ Equation tangent $: x+y+1=0$

It is tangent to circle $x^{2}+y^{2}=c^{2}$

$\Rightarrow c=\frac{1}{\sqrt{2}}$