Both laws are same and equivalent. If you have a triangle, then we can use the triangle law. If you have a parallelogram, use that law.
In general the parallelogram law is used, because it is easier to understand and apply.
If you have a parallelogram ABCD, then vector AB + vector BC = vector AC. AC is the third side of triangle ABC. This is triangle law.
vector AD + vector AB = vector AC.
this is parallelogram law.
(but we have vector AD = vector BC..)