I find the nicest way is to use cross product. This avoids any potential division by zero.

Let $$$L_1$$$ be defined by points $$$P_1$$$ and $$$P_2$$$ and $$$L_2$$$ be defined by points $$$P_3$$$ and $$$P_4$$$. Then $$$(P_2 - P_1) \times (P_4 - P_3) = 0$$$ if and only if $$$L_1$$$ and $$$L_2$$$ are parallel. Of course, when using floating points you would rather check if the absolute value of the cross product is less than some very small number, probably something like $$$\epsilon = 10^{-9}$$$.