a Quadrellateral
otherwise known as: a quadrilateral with two pairs of pairs of parallel sides.
Yes, by definition.
Any four-sided plane figure with two pairs of parallel sides is a parallelogram. Squares fit that description.
A parallelogram does not need to have any right angles at all. It is a 4 sides figure with 2 pair of parallel sides, or you can define it as a quadrilateral with 2 pair of parallel sides. So a square certainly is a parallelogram, but most parallelograms are not squares. A rhombus, a rectangle, and a square are all types of parallelograms.