= Solution
A smooth <vector field along a map> $\gamma:[a,b]\to M$ is a smooth section of the <pullback tangent bundle> $\gamma^*TM$: it assigns $V(t)\in T_{\gamma(t)}M$, with smooth coefficients in each local frame. It need not be the restriction of one ambient <vector field>, since a curve may return to the same point with different values of $V$.
An <affine connection>, also called a <Koszul connection>, induces a <pullback connection> on this bundle. In coordinates, write $V=V^i(t)\partial_i$ and $\nabla_{\partial_j}\partial_k=\Gamma^i{}_{jk}\partial_i$. Its <covariant derivative along a curve> is
$$
D_tV=\left(\dot V^i+\Gamma^i{}_{jk}(\gamma(t))\dot\gamma^jV^k\right)\partial_i.
$$
The connection transformation law, or the <Leibniz rule> applied to a change of frame, makes this expression independent of the frame. In particular it is defined even when $\dot\gamma=0$; it differentiates the section's coefficients as well as the frame.
The field is parallel precisely when $D_tV=0$. In a frame along a coordinate segment this is the <linear ordinary differential equation>
$$
\dot V^i=-A^i{}_k(t)V^k,\qquad
A^i{}_k(t)=\Gamma^i{}_{jk}(\gamma(t))\dot\gamma^j(t).
$$
The <Picard-Lindelöf theorem> gives a unique solution with prescribed initial value. Smooth coefficients on compact subintervals are bounded, and the usual norm estimate for a <linear ordinary differential equation> prevents finite-time blowup there. A finite sequence of coordinate segments covers the image of the compact interval $[a,b]$; solve successively and use uniqueness on overlaps. This gives a unique parallel field on the whole interval for every $V_a\in T_{\gamma(a)}M$.
Define <parallel transport> by $\tau_t(V_a)=V(t)$. Linearity of the equation and uniqueness give $\tau_t(cV_a+dW_a)=c\tau_t(V_a)+d\tau_t(W_a)$. Solving along the reversed curve supplies its inverse. Therefore \b[each $\tau_t:T_{\gamma(a)}M\to T_{\gamma(t)}M$ is a linear isomorphism].
For two <affine connections>, put $\Delta(X,Y)=\nabla_XY-\widetilde\nabla_XY$. Their rules give
$$
\Delta(fX,Y)=f\Delta(X,Y),\qquad
\Delta(X,fY)=f\Delta(X,Y),
$$
since the two $X(f)Y$ terms cancel. The <tensoriality> of this operation proves that the <difference of affine connections is a tensor>, of type $(1,2)$, with smooth coordinate coefficients $\Gamma^i{}_{jk}-\widetilde\Gamma^i{}_{jk}$.
A parametrized <geodesic> satisfies $D_t\dot\gamma=0$, or the <geodesic equation>
$$
\ddot\gamma^i+\Gamma^i{}_{jk}(\gamma)\dot\gamma^j\dot\gamma^k=0.
$$
With $v^i=\dot\gamma^i$, this becomes the first-order system $\dot\gamma^i=v^i$, $\dot v^i=-\Gamma^i{}_{jk}(\gamma)v^jv^k$. Its right side is smooth, so the <Picard-Lindelöf theorem> gives a unique local solution for each initial point and tangent vector. This establishes uniqueness with the parametrization fixed.
The two accelerations differ by
$$
D_t\dot\gamma-\widetilde D_t\dot\gamma=\Delta(\dot\gamma,\dot\gamma).
$$
If $\Delta(v,v)=0$ for every tangent vector, either acceleration vanishes exactly when the other does. Conversely, start the $\nabla$-<geodesic> with arbitrary initial vector $v$ at an arbitrary point. If it is also a $\widetilde\nabla$-<geodesic> with the same parameter, evaluating this identity initially gives $\Delta(v,v)=0$. Hence
$$
\boxed{\text{same parametrized geodesics}\quad\Longleftrightarrow\quad
\Delta(v,v)=0\text{ for every }v}.
$$
Polarization makes the latter condition equivalent to $\Delta(u,v)+\Delta(v,u)=0$: <parametrized geodesics determine the symmetric part of an affine connection>. In particular <torsion-free connections are determined by their parametrized geodesics>, since their difference is also symmetric and must then vanish.
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