I've just been playing around with the simplest notion of a connection on $\mathbb{R}^n$, that is
$$(\nabla_{v}X)^i=v(X^i)$$
with $X$ a vector field and $v$ a tangent vector at $p$.
From the definition of parallel transport, I know that we should regard $v$ as parallel to $X$ at $p$ iff $\nabla_vX=0$. I can't quite see how this relates to the notion of parallelism in $\mathbb{R}^n$ though!
I know that $v(X^i)$ tells me the rate of change of the component function $X^i$ in the direction of $v$ at $p$. I've convinced myself that the geometrical consequences of $v(X^i) = 0$ are the following.
In an infinitesimal neighbourhood of $p$ the vector field $X$ remains the same as you transport it along the straight line defined by $v$. Then for $v$ an arbitrary vector field we have that $X$ is a parallel vector field along the integral curves of $v$.
Are these the strongest things we can say? Many thanks!
Parallel in the old fashion Euclidean sense and parallel in the Riemannian geometry sense have little to do with one another.
In $\mathbb{R}^2$, consider the vector field which always points right and has unit length. That is, $v_{(x,y)} = (1,0)$ at every point $p\in \mathbb{R}^2$.
First, let $X_{(x,y)} = (e^x , 0)$. This is a vector field which always points right but as you get larger $x$ values, the arrows get longer. In the classical geometry sense, $X$ and $v$ are parallel at every point. However, if you compute, you'll see that $\nabla_v X \neq 0$, so these are not parallel in the Riemannian geometry sense.
Second, let $X_{(x,y)} = (0,1)$. This is a vector field which always points up with length $1$. In the classical geometry sense, since $v$ points right and $X$ points up, there is no way they are parallel anywhere.
Nonetheless, $\nabla_v(X) = 0$ so they are parallel in the Riemannian geometry sense.
The idea of Riemannian parallelism is that, to say $X$ is parallel along $v$ should mean that as you move along $v$, $X$ doesn't change. In the first example, $X$ does change as you move along $v$, while in the second example, it doesn't.