In geometry, two lines are said to be parallel if and only if they have no common points. However, there are many geometric theories in existence: Euclidean, Projective, Hyperbolic, etc. They differ by the sets of axioms at their foundations. In Euclidean Geometry, parallel lines stay on the “same distance from each other” whereas in other geometrics this is not so.
Therefore, in usual geometry, parallel lines do not meet and there is no such thing as infinity and it is wrong to say that parallel lines meet at infinity. In other geometric systems, parallel lines may meet at a point at infinity. Whether this is one single point or different points for different classes of parallel lines, depends on the particular geometric system that is considered.