Equation of the line passing through (1, 1, 1) and parallel to the plane 2x + 3y + z + 5 = 0, is

$\frac{x-1}{-1}=\frac{y-1}{1}=\frac{z-1}{-1}$

Clearly, all lines in the given options pass through (1, 1, 1). We know that a line is parallel to a plane if the normal to the plane is perpendicular to the line.

$\vec{n}= 2\hat{i} + 3\hat{j} + \hat{k}$ and the line in option (b) is parallel to the vector $\vec{b}= - \hat{i}+\hat{j}-\hat{k}$ such that 

$\vec{b} .\vec{n} = - 2+3 -1 = 0 $ i.e, $\vec{b}⊥ \vec{n}.$

Hence, option (b) is correct.