A parallelogram's definition is a quadrilateral with two sets of parallel lines. This means lines MA and TH are parallel, and AT and MH are parallel. MA's slope is (2-(-3)/(-4-(-1)), or -5/3. TH's slope is (3-8)/(9-6) or -5/3 too. This means they are parallel, as they have the same slope. Do the same with AT and MH, and you get their slopes as 3/5 and 3/5. This means that the whole shape is a quadrilateral.
For the 2nd part, rectangles have sides that meet at right angles, which means that they are perpendicular, and perpendicular lines have slopes that are the negative reciprocals of each other. -5/3 and 3/5 are negative reciprocals of each other, so therefore it is a rectangle.