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No, a circle can't be a parallelogram. A circle is a curve. A parallelogram is a quadrilateral with two pairs of parallel sides constructed with four line segments. The line segments are straight, and the circle is a continuous curve.
There are more than two: A parallelogram has two sets of parallel line segments which may be of unequal length; A rectangle is a parallelogram that has line segments that meet at right angles; A rhombus has to sets of parallel line segments which are all of the same length; A square is a rhombus which has line segments meeting at right angles, or stated differently a square is a rectangle which has all line segments of the same length.
Triangle diamonds and parallelogram
Draw two line segments forming an angle. From the open end of one of the line segments draw another line segment that is on the same side of the line as the other arm of the angle but not parallel to it. Draw the fourth side. You will have a quadrangle that is not a parallelogram.
Technically, lines are continuous, so parallelograms are actually composed of line segments. The line segments composing the sides of the parallelogram, come in two pairs in 2D space. Each pair is composed of two line segments that are parallel to each other, but do not occupy the same line. The two pares of line segments must all meet to then form a quadrilateral.