Why are parallel vector fields called parallel?

In Lee's "Riemannian Manifolds: An Introduction to Curvature" given a curve $\gamma:[a,b]\to M$ and a tangent vector $V_0\in T_{\gamma(t_0)}M$, where $t_0\in [a,b]$, there is a drawing of the parallel translate of $V_0$ in figure 4.7. This parallel vector field seems to be drawn so that it is perpendicular to $\gamma$ at every point.

Is there a geometric interpretation as to why parallel vector fields are called parallel? Why is the parallel translate drawn in this way?

A vector field $X$ along a curve $\alpha$ is parallel if $$\nabla_TX=0$$ This equation means that the vector field $X$ does change along $\alpha$. Geometrically, all values of $X$ along $\alpha$ seems to be parallel.