A parallelogram is defined by having its opposite edges equal and parallel: that is, the same length and direction. So as direction (or displacement) vectors they are equal (vectors are equal if and only if they have the same direction and magnitude). If the position vectors of the vertices are a,b,c,d then the direction vectors representing the edges are b-a = d-c and c-b = d-a. You can check that this is indeed true for your four points.
Incidentally you can see that b-a = d-c => b+c = a+d => c-b = d-a and vice versa. So you only need to check one of these three equalities. From the middle equality here you also derive the condition (b+c)/2 = (a+d)/2: that is, a quadrilateral is a parallelogram if and only if the mid-points of the two diagonals coincide.