How do you explain how a parallelograms' angles add up to 360 degrees?

Draw a diagonal. The result is two triangles. The sum of the angles in any planar triangle is 180o. Twice 180 = 360.

^That is not really answering the question.

In a parallelogram the same side angles are supplementary (any side) and that makes 180o do this on the other side, add the two 180o and you have 360o