Quadrilateral can certainly be parallelograms, but they can also be squares, rectangles, and other 4-sided shapes. All parallelograms are quadrilaterals but not all quadrilaterals are parallelograms.

No. A rectangle and a parallelograms are desciptions of quadrilateral shapes. There is no indication of the size of either. So some rectangles are smaller than some parallelograms and some parallelograms are smaller than some rectangles.