Example 4 Suppose the base of the pyramid $P-ABCD$ is not a parallelogram, and a plane $\alpha$ cuts this pyramid such that the section is a parallelogram. Then the number of such planes $\alpha$ ( ).
(A) does not exist
(B) is only one
(C) is exactly two
(D) is infinitely many
Explanation: As shown in Figure 5, extend \( BA \) and \( CD \) to intersect at point \( M \), and connect \( PM \). Then \( PM \) is the intersection line of the lateral faces \( PAB \) and \( PCD \). Similarly, \( PN \) is the intersection line of the lateral faces \( PAD \) and \( PBC \). Let the plane determined by the lines \( PM \) and \( PN \) be \( \beta \). Construct a plane \( \alpha \) parallel to plane \( \beta \) that intersects the lateral faces of the pyramid, then the resulting quadrilateral is a parallelogram (the quadrilateral \( A_1B_1C_1D_1 \) in Figure 5). It is easy to see that there are infinitely many such planes \( \alpha \). Therefore, the correct choice is (D).