Let the equation of plane parallel between $2 x-2 y+z+3=0$
and $2 x-2 y+z+\frac{9}{2}=0 $ is
$2 x-2 y+z+c=0$
Now, distance between two given parallel lines
$=\frac{\frac{9}{2}-3}{\sqrt{2^{2}+2^{2}+1^{2}}}$
$=\frac{3 / 2}{3}=\frac{1}{2}$
$\because$ Required plane is equidistant from given planes
$\therefore \frac{c-3}{\sqrt{2^{2}+2^{2}+1^{2}}}=\frac{1 / 2}{2}$
$\Rightarrow \frac{c-3}{3}=\frac{1}{4}$
$\Rightarrow c=\frac{3}{4}+3$
$\Rightarrow c=\frac{15}{4}$
$\therefore $ Required equation of plane is
$\therefore 2 x-2 y+z+\frac{15}{4}=0$
$\Rightarrow 8 x-8 y+4 z+15=0$