Can a quadrilateral be both a parallelogram and a rhombus?

Yes. A quadrilateral is a 4 sided shape with 2 diagonals and all angles add up to 360

It is also a parallelogram, since it has two pairs of parallel sides. A square has two pairs of parallel sides, four right angles, and all four sides are equal. It is also a rectangle and a parallelogram. A rhombus is defined as a parallelogram with four equal sides.