If the lines are exactly parallel, they never meet, of course. If they are not exactly parallel, that is, with a small angle between them, then they can meet.
Light rays from the sun are clearly not exactly parallel, because they do meet at the sun!
Have you tried my little experiment yet?
Also note that even lines that are exactly parallel can appear to meet "at infinity" because of geometric perspective effects. Have you ever looked down a long straight road or railroad track that extends to the horizon on a flat plain? The sides of the road or the rails of the track appear to meet at the horizon.