I'm self-studying differential geometry and general relativity - so far I know of smooth manifolds, vector fields, smooth curves on manifolds, and basics of connections. I know that a smooth vector field $X$ is parallel w.r.t. a connection $\nabla$ if $\nabla X$ vanishes. My interpretation of this is that $\nabla X(\omega,Y)=0$ for all smooth covector and vector fields $\omega,Y$. i.e. $\nabla_YX=0$ for all smooth vector fields $Y$.
I'm also reading General Relativity by Wald, in which I found a statement: "A space will be curved if and only if some initially parallel geodesics fail to remain parallel."
I tried to search for "parallel curves on manifold" on google, but didn't really find anything concrete on how to identify if two smooth curves on a manifold are parallel. The results are only related to "parallel vector fields", "parallel transport", etc. So I want to ask here: is there a well-defined notion of two smooth curves on a smooth manifold being parallel? If so, what is it? e.g. what makes us say that longitude lines on the Earth are parallel? Or how do we prove that latitude lines are or aren't parallel?
This vague statement is merely a comment in the introduction to the chapter on curvature. You just need to keep reading. I recall later on Wald introduces the notion of parallel transport, and the Jacobi equation. The relationship between parallel-transport and curvature, and also Jacobi's equation is also discussed in any Riemannian geometry text, e.g Lee's. On MSE itself, see this and this.
Anyway, since you're so eager, let me briefly outline how Jacobi's equation (not to be confused with the Jacobi identity) gives some nice qualitative insight to the interpretation curvature. The setup is as follows: start with a smooth manifold $M$, a torsion-free connection $\nabla$ on the tangent bundle $TM$, and let $I,J\subset\Bbb{R}$ be open intervals around the origin, and suppose you have a smooth mapping $f:I\times J\to M$ such that for each $s\in J$, $f(\cdot, s)$ is an affinely parametrized geodesic (also called autoparallel since we're being general and not assuming a metric-compatible connection) in $M$. We think of the parameter $s\in J$ as the 'variation parameter', and $t\in I$ as the 'curve parameter' (they're also called the transverse and longitudinal parameters respectively). The mapping $f$ gives rise to two vector fields along $f$, which by common abuse of notation, I shall denote as $T=\frac{\partial f}{\partial t}$ and $S=\frac{\partial f}{\partial s}$ (really I mean $Tf\left(\frac{\partial}{\partial t}\right)$ and $Tf\left(\frac{\partial}{\partial s}\right)$ respectively). It is now a reasonably simple exercise (and also proved in the texts) that the vector field $S$ obeys the following equation along the mapping $f$ (here, torsion-freeness is used): \begin{align} \nabla_T\nabla_TS&=R(T,S)T. \end{align} Here, $R$ is the Riemann curvature endomorphism (i.e the full $(1,3)$ tensor field). This is known as the Jacobi equation, or the geodesic-deviation equation. So, we have the three vector fields
Each of these vector fields tells us how the nearby geodesics (i.e nearby $s$-values) behave as we let their curve parameter $t$ run along. To get a rough pictorial idea, draw a family of geodesics on the sphere, and try to visualize these vector fields. The statement Wald is making is the following:
The necessity is obvious from the Jacobi equation. The sufficiency also follows from the Jacobi equation, with a little more algebraic finesse: the hypothesis implies $R(T,S)T=0$ for all possible $T,S$. In particular, this implies that for all $p\in M$ and all $u,v\in T_pM$, we have $R_p(u,v)u=0$, i.e for all $u,v,w\in T_pM$, we have $R_p(u,v)w=-R_p(w,v)u$. Combining this with the skew-symmetry $R_p(u,v)=-R_p(v,u)$ and the algebraic Bianchi identity (which holds since we have a torsion-free connection) gives (I omit the point $p$ for convenience in notation) \begin{align} 0&=R(u,v)w+R(v,w)u+R(w,u)v\\ &=R(u,v)w+[-R(u,w)v]+R(w,u)v\\ &=R(u,v)w+2R(w,u)v\\ &=R(u,v)w-2R(v,u)w\\ &=3R(u,v)w, \end{align} and hence $R(u,v)w=0$, proving that the Riemann curvature vanishes identically.
So, what this is saying is that if the Riemann curvature is non-zero, then it is possible to find a family of geodesics $f$, such that when $t=0$, the vector field $\nabla_TS$ vanishes 'initially, nearby geodesics are parallel', but that at some later time $t$, $\nabla_TS$ is non-zero 'they fail to remain parallel' (so necessarily $\nabla_T\nabla_TS$ is non-zero). If you want pictures (some good, but some too imprecise), see MTW's tome