A unit vector parallel to the points from (a,b,c) to (d,e,f) is just this:
v1 = [a,b,c];
v2 = [d,e,f];
u = v2 - v1;
u = u/norm(u);
A point on a parallel line that goes through (l,m,n) is then just this:
p = [l,m,n];
x = any scalar value
point_on_line = p + x * u;
