You don't need to prove it. A parallelogram is defined that way. If they are not parallel, it is not a parallelogram.

Suppose in a parallelogram ,the sides are given A B C D so we'll divide it into 2 triangles ABC ad ADC so in triangles ABC ad ADC,we have AB||DC i.e AB=DC AC=AC (common) then AD||BC so,AD=BC now, AB||DC and AD||BC..therefore all sides are parallel in a parallelogram.