Triangles can't have parallel lines, at least not in the type of math you're dealing with.
You could argue that a triangle on a sphere can have lines that are parallel at certain points, but these lines won't be parallel throughout the triangle. Consider a triangle (northern hemisphere) formed by the Earth's equator, the 0 degree longitude line, and the 90 degree longitude line. Assume for the moment that the Earth is a perfect sphere (it isn't in real life). The 0 degree line and 90 degree line are parallel to each other at the points where they hit the equator, but they are perpendicular to each other where they hit the north pole.