The equation of the plane through the line $x + y + z + 3 = 0 = 2x - y + 3z + 1 $ and parallel to the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$, is

$x- 5y + 3z = 7 $

The equation of the plane through the given line is 

$x + y + z + 3 + λ(2x - y + 3z + 1) = 0 $

$⇒ x(2λ + 1)+ y(1 - λ) + z(3λ+1)+λ + 3 = 0 $

It is parallel to the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$

$∴(2λ+1)+2(1-λ)+3(3λ+1)=0 ⇒ λ = -\frac{2}{3}$

Putting $λ =-\frac{2}{3}$ in (i), we obtain $x - 5y + 3z = 7 $ as the required equation.