A quadrangle with two pairs of equal sides is a square.
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That's not possible. No rhombus (quadrilateral with two pairs of equal sides) cannot be a parallelogram.
There is no specific name for a quadrangle with two equal sides.
It could look like an arrowhead (or delta).
A Kite. The two sets of equal sides are consecutive.
The figure, as described, is impossible. If a quadrangle has 2 pairs of parallel sides then the number of right angles can be 0 or 4, but not 2.