Okay, not parallel lines but parallel geodesic curves. I don't know if I can ELI5 geodesics, but I'll have a go

Okay, you can't take a straight line on a sphere, obviously. But if you walked around the equator, most people would a agree you'd walked pretty much a straight path. However, if you walked a 1 meter circle around the North pole, no-one would recognise that as a straight path, even though they're both lines of latitude and they're both parallel

What's the difference? Pick any two points on the equator. The shortest path between them (staying on the sphere), also follows the equator. For the small circle you don't have that property - you can find a shorter curve that cuts through the interior of the circle. Curves that have this shortest distance property are called geodesics

So the statement about flatness should be; two geodesics - that is, two shortest distance curves - that are parallel at some point, stay parallel their whole lengths