> Unfortunately, of course, our familiar space is not Euclidian. Well, mine isn't.
And? Mathematics is in no way related with the real world. The fact that some mathematical structures can be used to model objects and behaviors from the real world is just a happy coincidence.
> But, given a nicely-behaved Euclidian space, the challenge would be to prove that there can be at most one line through a point that is not on a second, parallel line.
So you have the following choices:
1) Leave it out entirely. You'll have much poorer set of theorems.
2) Use it. You'll get boring old geometry.
3) Replace it with another axiom. You'll get non-Euclidean geometries as a result.