By definition as in DoCarmo’s Riemannian Geometry book, in a smooth manifold $M$ a vector field $V$ along a smooth curve $c:(a,b)\rightarrow M$ is a mapping $t\mapsto V(t)\in T_{c(t)}M$ such that $t\mapsto V(t)f$ is a smooth map for all smooth maps $f\in C^{\infty}(M)$.
Suppose that $c_1,c_2$ are two smooth curves in $M$ such that $c_1(t_0)=c_2(t_0)$, and $c_1(t_1)=c_2(t_1)$ where $t_0$ and $t_1$ are two distinct points in the (same) domain of $c_1,c_2$. Suppose that $V_0\in T_{c_1(t_0)}M$. There exists a unique parallel vector field $V_1$ along $c_1$ such that $V_1(t_0)=V_0$, and similarly a unique $V_2$ along $c_2$ such that $V_2(t_0)=V_0$. Does it follow that $V_1(t_1)=V_2(t_1)$?
Not necessarily, unless the space is flat. This is actually a definition; a space is flat if and only if the property you ask about holds. The Euclidean space is flat.
These things are the starting point of the concept of curvature.
The following picture is taken from Wikipedia and it shows how the choice of a path can produce different parallel trasported vectors, in the case of the sphere.