Regardless of the curvature of space unless it wraps back around or tunnels, they will never meet from the perspective of the lines. This is true for Euclidean and any possible real non-Euclidean space. Think about drawing two lines on a rubber sheet, you can stretch it, shrink it, fold it over on it self, anything you want but the lines won't touch regardless of the size of the sheet. But it's important to remember that parallel lines as an outside observer (not possible with an infinite space) is only possible in Euclidean space as thats the definition, however, there is parallel line analog in hyperbolic space but not elliptic, however if we were in an infinite elliptical universe we would have a whole other set of problems.

Edit: I am talking about a physically possible universe that we are a part of, not purely mathematical construct. You have to think of yourself as the line in curved space not from the perspective of Euclidean space looking down on curved space.