What type of plate boundary is parallel to the direction of the plate movement?

Would that be TECTONIC..

Doubt it. "Tectonic' is a term related to the study of earth crust movement, and the results thereof. No direction is implied.

A transform fault (http://en.wikipedia.org/wiki/Transform_fault).

Also consult this discussion of the types of plate boundaries.
Plate boundaries (http://scign.jpl.nasa.gov/learn/plate4.htm)

Yep transform fault.