It's best not to think "parallel", since all vectors are considered to have their base at the origin. If you have a set of two nonzero vectors {a,b}, then the following are equivalent:

- a and be are linearly dependent,
- a=cb for some scalar c.

If you have more than two vectors, then you can have linear dependence without any pair being linearly dependent among themselves. For instance, the set of vectors {(1,0), (0,1), (1,1)} is linearly dependent in R^2, although none of them is a scalar multiple of any other.