A square is both a quadrangle and a parallelogram.
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a square * * * * * It need not be a square. It is simply a right angled quadrangle! It can be a trapezium, kite, arrowhead, rectangle or square. It cannot be a parallelogram or rhombus - other than a rectangle or square respectively.
If all four sides are equal, it must be a rhombus, which is a parallelogram. The square is a special case of a rhombus.
trapezoid
The answer is: quadrangle
I do not believe that such a quadrangle can exist in Euclidean space.