If by parallel, you mean two lines that do not intersect, yes, it is possible to draw them on the surface of a sphere. They will end up being circles, and most pairs will not be equal in size. If you add the idea that the two lines also continue to infinity to the definition, then you cannot draw such things on the surface of a sphere.
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Draw one pair of intersecting (crossing) lines. For each of the two lines, draw another that is parallel to it. The four lines will enclose a space which is a parallelogram.
Draw a line. Draw a perpendicular to that line then a perpendicular from that one.
It is impossible to have a triangle with 3 right angles. It is possible to draw a triangle with three right angles on the surface of a sphere: www.metacafe.com/watch/769025/270_degree_triangle_yes_3_right_angles
Draw two parallel lines of unequal length, and connect their end points. If you have a right angle, it is a right trapezoid. If the non-parallel sides are equal in length, it is an isosceles trapezoid.
To draw a square with 3 lines, you can draw two perpendicular lines to form a right angle. Then, draw a diagonal line connecting the endpoints of the two perpendicular lines. This diagonal line will complete the square shape with only 3 lines.