Difference between triangle law and triangle law of vector addition
These both laws are the same. But there is a minor difference between them on the basis of theory, application, & usefulness.
Parallelogram Law: If two vectors, vector 'a' & vector 'b' are represented in magnitude & direction by the two adjacent sides of a parallelogram, then their sum, vector 'c' is represented by the diagonal of the parallelogram which is coinitial with the given vectors.
Triangle Law: If two vectors are represented in magnitude & direction by the two sides of a triangle taken in the same order, then their sum is represented by the third side taken in the reverse order.
The parallelogram law asks to put the tails (end without the arrow) of the two vectors at the same point.
The triangle law asks to take the tail of the second vector and place it at the head of the first vector.
According to the question put by you, parallelogram method is more useful as the vectors already have a common base, i.e the origin of the Cartesian coordinate system.
Moreover, if you know the algebraic method of addition of two vectors, then that method dominates all the graphical methods!