Was it ever proven that parallel lines on a surface with zero curvature never meet?
It’s not proven, it’s postulated. Or to put it another way, if there are two lines that never meet, we define those lines as “parallel”. If the lines do meet, then they aren’t parallel.
It’s possible to have a mathematics that doesn’t have the Parallel Postulate, that’s the famous “non-Euclidean” geometry people sometimes talk about.
Which means you cannot prove the parallel postulate (axiom) from the other postulates of Euclidean geometry: If you could, it would be a logical consequence of them, and forced to be true given the rest of the axiom system, but, since you can have a perfectly consistent axiom system by taking the other axioms and negating that one, it must not be a consequence of the rest of them.
This doesn’t stop people from occasionally claiming they’ve proven it, mind you, it just makes them angry when the rest of the mathematical world dismisses their claims out of hand.
It is not a postulate or axiom but simply a definition:
(Book 1, Def. 23)
ETA do not confuse this with Euclid’s fifth postulate or Playfair’s parallel postulate.
Alternately, if by “parallel lines”, you mean lines which make the same angle with a line that intersects both, then those lines never meeting is what the Parallel Postulate says. But then we get to the other part of your question, “on a surface with zero curvature”, and a surface with zero curvature is defined to be one where Euclid’s Parallel Postulate holds.
I would disagree with many of the above posts, because clearly not all of Euclid’s postulates apply to all zero-curvature surfaces, e.g. not all straight lines are infinitely extendable on a flat torus, so it seems fair to ask whether the parallel postulate applies to general surfaces with constant, zero curvature. In fact it is easy to find an example where the parallel postulate almost fails: parallel lines meet at the apex of a cone which is flat everywhere except for the apex (a conical singularity occurs at the apex, so you would need to go beyond Riemannian manifolds to describe the apex).
I’m not entirely sure that it has been proven generally for non-compact higher-dimensional manifolds of constant, zero curvature, but I could be wrong.
When you say that parallel lines meet at the apex of a cone, what do you mean by “parallel”?
The geometry of a flat torus is, of course, not Euclidean: for instance the axiom that given three points on a line, no more than one lies between the other two, is violated.
A cone is rather like a cylinder if the apex is at infinity, if you see what I mean. Neither are globally Euclidean, for the same reason as above.
Whether two lines on a cone are parallel can be defined from the isometry to a portion of the Euclidean plane.
I’ve told you and told you and told you. Never ask a question about math. Never. Especially a simple one. You won’t like the answers. But does anybody listen to me? Noooooooooo. Someone asks one and suddenly non-compact higher-dimensional manifolds are introduced and everyone’s running screaming for the door and there’s grade juice everywhere. And then what happens? Somebody asks another math question.
Who’s going to clean up all this grape juice, I ask, but nobody listens to that either. It’s enough to drive a person non-Euclidian.
Actually, what is the question concerning manifolds, conifolds and/or Riemannian manifolds? The OP mentioned curvature so we must be in the realm of differential geometry.
Exactly. Open Pandora’s box, and stuff flies out.
My evil deed for the day. 
What if the parallel lines are on a treadmill?
Depends. Are they frictionless?
I remember a BC cartoon where Peter is chastising the group with, “You oafs can’t grasp the concept of parallel lines not meeting so I shall have to prove it to you.” He presses a long, straight branch with a Y at the end, like a long-handled slingshot, against the ground, says, “See ya!” and walks off. The rest of the Sunday strip has him walking night and day with the stick scratching twin furrows in the dirt – and the two arms are wearing down. Just as he gets back to the rest, the arms are gone and the lines do meet, to his chagrin.
Fear not: Great Cthulhu will eat you first.
Hart did a few on parallel lines. In one strip he drags two sticks down the beach leaving parallel lines. Eventually he turns around and sees the lines converging in the distance where he started. He walks back and see they’re apart again, but converge in the other direction.
I am a laymen of math laymen-laymen but isn’t the very definition of “parallel,” two lines that never meet? If they could meet, then they wouldn’t be “parallel.”
This sounds akin to “What if a right triangle did not have any angle that was 90 degrees?” Then it wouldn’t be a right triangle.
Yes, and no. See for example the discussion in posts 5, 6, 7, & 9.
It is not a given that a “right” angle must measure 90 degrees. It depends upon your geometry and your measuring methodology. To some extent, that question is identical to the one being presented.