Smoothly move player over a collection of checkpoints

I have a collection of vector3 points in an array. Currently my player is experiencing a jerk while moving (after reaching each point in an array it stops and start its movement). Below is my logic for movement. How do i make the movement smooth?

 void Update ()
         {                
                 if (isMoving)
                         MoveToTargetPoint ();
         }
 
 void MoveToTargetPoint ()
         {
        // if player has  reached the threshold distance of each checkpoint change the target to next checkpoint
                 if (index <= path.Length - 1 && Vector3.Distance (target, parentT.position) <= 0.3f) {
                         target = path [index++];
                         movementTime = 0f;
                 } 
 //if player has reached the last checkpoint then stop moving
 else if (index > path.Length - 1 && Vector3.Distance (target, parentT.position) <= 0) {
 
                         isMoving = false;                        
                 }
 void FixedUpdate ()
         {
                 if (isMoving) {
                         movementTime += Time.deltaTime * 0.8f;
                         parentT.position = Vector3.Lerp (parentT.position, target, movementTime);
                 }
         }
 
                 
         }