In this note we show that $B$-scrolls over null curves in a 3-dimensional Lorentzian space form $\bar{M}^3_1(c)$ are characterized as the only ruled surfaces with null rulings whose Gauss maps $G$ satisfy the condition $\Delta G=\Lambda G$, $\Lambda \:{X}(\bar{M})\rightarrow {X}(\bar{M})$ being a parallel endomorphism of ${X}(\bar{M})$.