Example 11 Divide the sides of an equilateral triangle $ABC$ into $n$ equal parts, and draw lines parallel to the sides through each division point within the triangle, thus forming a grid of equilateral triangles. Find the number of parallelograms in the equilateral triangle.
Solution: First, as shown in Figure $30-4$, consider parallelograms whose sides are not parallel to $B C$. Clearly, such a parallelogram has its two pairs of opposite sides extended to intersect $B^{\prime} C^{\prime}$ at four points $i, j, k, l$; conversely, any four points on $B^{\prime} C^{\prime}$ correspond to one such parallelogram. Therefore, there is a 1-1 correspondence between the set of parallelograms whose sides are not parallel to $B C$ and the set of all combinations of four points on $B^{\prime} C^{\prime}$. Thus, the number of parallelograms whose sides are not parallel to $B C$ is $C_{n+2}^{4}$. By symmetry, the total number of parallelograms in the triangular grid is $3 C_{n+2}^{4}$.