"In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure." (bold mine) -- Wikipedia
So, you have four points. There are four sides (this is a quadrilateral). The sides have four distances (and you may use the distance formula to find those values). If there are two pairs of equal distances, then you have shown that these four points form a parallelogram.
p.s., Often, the key to solving problems involves knowing and applying definitions, formulas, procedures, and understanding the question being asked. Getting the correct answer is not the goal -- understanding the process is the goal (understanding is useful next time, the correct answer to this problem is not).