Understanding the Problem
To determine the number of parallelograms formed by two sets of parallel lines, we need to understand how these lines intersect.

Components of the Problem
- **First Set of Lines**: 5 parallel lines
- **Second Set of Lines**: 3 parallel lines

Forming a Parallelogram
A parallelogram is formed by selecting two lines from the first set and two lines from the second set. The selected lines from each set must be non-parallel to each other to ensure the shape is a parallelogram.

Calculating Combinations
1. **Selecting Lines from the First Set**:
- We can choose 2 lines from 5 parallel lines.
- The number of ways to choose 2 lines from 5 is given by the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).
\[
C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\]
2. **Selecting Lines from the Second Set**:
- We can choose 2 lines from 3 parallel lines.
\[
C(3, 2) = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3
\]

Total Number of Parallelograms
To find the total number of parallelograms, we multiply the combinations from both sets:
\[
\text{Total Parallelograms} = C(5, 2) \times C(3, 2) = 10 \times 3 = 30
\]

Final Answer
Thus, the total number of parallelograms that can be formed from the given sets of lines is **30**.