A triangular prism is one that has 2 triangular-shaped bases with three sides (called lateral faces) that are rectangles.
By taking a shape and making a "cut" straight across it, we create a cross-section (see the attached picture).
The cross-section of a triangular prism then would be a triangle. Is a triangle a parallelogram? A parallelogram is a four-sided figure that has opposite sides that are parallel and are equal in length. A triangle is not a parallelogram because it does not have four sides, nor parallel sides. Since the cross-section of a triangular prism is a triangle (which cannot be a parallelogram), a parallelogram can never be the cross-section of a triangular prism.