A quadrilateral (and other polygons) is inscribed in a circle if all its vertices lie on the circle.
1) If the diagonals of the parallelogram pass through the center of the circle, and diagonals are also congruent and perpendicular to each other, then the parallelogram inscribed in the circle is square or rhombus.
2) The other parallelogram that can be inscribed in a circle is rectangle but its diagonals are not perpendicular to each other.