Draw a circle, and a inscribed quadrilateral ABCD with 4 sides of different length in the circle. A, B, C & D are on the circumference. Connect A to C & B to D. Since angles subtended at the circumference by the same cord are equal, so ∠ACB=∠ADB & ∠ACD=∠ABD. Here ∠ACB+∠ACD=∠BCD, so ∠BCD=∠ADB+∠ABD.
While, in △ABD, ∠BAD=180°－(∠ADB+∠ABD)=180°－∠BCD. Thus∠BAD+∠BCD=180°: ∠A+∠C=180°. Therfore the opposite angles of a inscribed quadrilateral ∠A & ∠C (or∠B & ∠D) are supplementary.
Meanwhile, in a parallelogram, the opposite angles are identical to each other. So, if the figure is a inscribed parallelogram, ∠A=∠C, and ∠A+∠C=2∠A=180°, so ∠A=∠C=90°. So the quadrilateral will be a rectangle or a square. Therefore a parallelogram that is not a rectangle or a square can NOT be inscribed in a circle.