4. Find a parallelogram such that its two adjacent sides are $a$ and $2a$, and the acute angle between the two diagonals is $60^{\circ}$. Such a parallelogram should be ( ).
(A) only one
(B) two
(C) infinitely many
(D) none at all
4. D.

As shown in Figure 4, intuitively, the locus of point $O$ for all $\triangle OBC$ with $BC=a$ as the base and the vertex angle $O$ as $60^{\circ}$ is the major arc $\overparen{BC}$ (as shown in Figure 4 (b)), excluding points $B$ and $C$. Among these, the maximum value of the median on the base $BC$ is the height $OE$ of the equilateral $\triangle OBC$.

And $OE=\frac{\sqrt{3}}{2} a < a$, which contradicts the $OE=a$ in the parallelogram shown in Figure 4 (a).
Therefore, such a parallelogram does not exist.