If the acceleration is parallel to the velocity vector. Then it should be perpendicular to the acceleration vector you mentioned in the beginning.
If you are talking about circular motion....
The perpendicular acceleration component to the velocity is known as the centripetal acceleration.
While the parallel component to the velocity is known as the tangential acceleration.
Use this image as a reference,
https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/222/2014/12/20103410/Figure_11_01_03a.jpg
Please comment below if you have further questions :)