A parallel plane P(x,y,z) must have the same normal vector (any scalar multiple but 0). If it is supposed to go through the origin, then P(0,0,0)=0.
If the equation for a plane is Ax +By +Cz + D = 0, then it is easy to find D so it passes through the origin.
so you can find a vector between any point on the line L and P, lets call it v2.
Now you have 2 vectors, v1 and v2, they lie in the plane containing P and L, so if you can find a vector perpendicular to both of them,
then you have a normal vector for the plane. Then it is easy to find D by substituting e.g. P in the resulting equation.
Hope that helps