If vectors are thought of as triplets of numbers (a,b,c) then why should these triplets be added by the parallelogram law? It all depends what these triplets are being used to represent.
If you start with addition of positive numbers on the number line, say 5+3, this can be seen as laying down two line segments of length 5 and 3 along a line starting at the origin. This can be generalized to line segments in a plane by allowing them to point in different directions.
If you start with two adjoined line segments then you can expand the drawing to a parallelogram and it is due to Euclidean geometry that opposite sides of a parallelogram have the same length and point in the same direction. So the parallelogram gives two different paths to move from the origin to the resulting point which is saying that the paralleogram law is equivalent to the commutative law A+B=B+A.
Even in one dimension, the line segments can be given a direction, so that 5+(-3) means putting down a line of length 5 in the +direction, and then turning around and drawing a line of length 3 in the opposite direction to arrive at the number 2. Being commutative you could also start with a line of length 3 in the negative direction from the origin along the real number line and then turn around and draw a line of length 5 in the positive direction again arriving at 2.
So the parallelogram law of vector addition is a straightforward extension of addition of ordinary numbers as directed line segments and is compatible with addition of negative numbers.
Long before cartesian coordinates were developed, people would have used geometry including the pythagorean theorem to calculate the addition of distances and velocities.
Then special relativity was discovered and it was found that the parallelogram law no longer holds for velocity addition, so to answer your question: other laws are indeed sometimes used. The question is: how do we know which laws apply to which aspects of nature? We don't know. We make observations and then make assumptions about how things behave based on those observations. We try to use mathematical structures that accurately describe the observed behaviour, and assume that if our models are accurate enough then any calculations derived from those models will also describe nature accurately.