If either  or  is a zero vector then it would be orthogonal to , but could not be parallel to the other vector.

Parallel vector : Vectors are parallel if they have the same direction.

Zero vector : It is not pointing in any particular direction, so it has an undefined direction.

Using the above statements of Parallel vector and Zero vector, justify that If and are both orthogonal to in the plane, it may or may not be parallel. 

Therefore, the statement is false.

The statement is false.