A parallelogram is formed by selecting two lines from one set and two lines from the other set. Each pair of lines from one set intersects with each pair from the other set, creating a parallelogram.

The number of ways to choose 2 lines from 4 is given by the combination formula: $({}^4C_2) = \frac{4 \times 3}{2 \times 1} = 6$. Since both sets have 4 lines, the number of ways to choose 2 lines from each set is 6.

Multiply the number of ways to choose 2 lines from the first set by the number of ways to choose 2 lines from the second set: $6 \times 6 = 36$

Therefore, the total number of parallelograms that can be formed is 36.