4. Let the base of the quadrilateral pyramid $P-ABCD$ not be a parallelogram. If a plane $\alpha$ cuts this quadrilateral pyramid such that the cross-section is a parallelogram, then the number of such planes $\alpha$ ( ).
(A) does not exist
(B) there is only 1
(C) there are exactly 4
(D) there are infinitely many
4. D.

Let the intersection lines of two sets of non-adjacent lateral faces of a quadrilateral pyramid be $m$ and $n$. The lines $m$ and $n$ determine the plane $\beta$. Construct a plane $\alpha$ parallel to plane $\beta$ that intersects the lateral faces of the quadrilateral pyramid. The resulting quadrilateral is a parallelogram. There are infinitely many such planes $\alpha$.