Maximum number of common normals of  $y^2=4ax,x^2=4by$  can be equal to 

Normal of  $y^2=4ax$  and $x^2=4by$   are
$y=mx-2am-a{m}^3$  and  $y=mx+2b+\frac{b}{m^3}$
Common normal is
 $2b+\frac{b}{m^2}-2am+a{m}^3=0$
 $a{m}^5+2a{m}^3+2b{m}^2+b=0$
There canbe atmost 5 common normals