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They can not be line segments on the same line, but they can both be line segments.
False. They can only be straight line segments: there cannot be any curved line segments.
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.
These are line segments that are the same in direction and length. * * * * * They need not be in the same direction.
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.