I‘m first going to comment on your two questions and then, for completeness, I‘ll give a sketch of proof for the exercise.
1) You are right that if we look at the curve as $t\to \gamma(e^{2\pi i t})$ then there is a parallel vector field $V(t)$ along $\gamma$ with $V(0)=v$ for every $v\in T_pM.$ However it might be that $V(1)\neq V(0).$ But as now we see the curve as a mapping $\gamma:S^1\to M$ we also want a parallel VF $V$ as a map from $S^1$ sucht that $V(t)\in T_{\gamma(t)}M$ and therefore the parallel VF should have the same value when starting and when „finishing the whole round“.
If this is not quite clear, then don‘t worry: In my answer below you can still think of the curve as a map $\gamma:[0,1]\to M,$ but we will see that there is a parallel vector field with $V(1)=V(0).$
2) Let $\omega$ be an orientation form for $M,$ i.e. a non-vanishing $n-$form (where $n$ is the dimension of $M$).
For each $t\in [0,1]$ we can define an orientation on the orthogonal complement of $\gamma‘(t)$ by declaring $v_1,...,v_{n-1}$ to be positively oriented if
\begin{equation}
\omega_{\gamma (t)}(v_1,...,v_{n-1},\gamma'(t))>0.
\end{equation}
Let $A:\left(\mathbb{R}\gamma'(0)\right)^{\perp}\to \left(\mathbb{R}\gamma'(0)\right)^{\perp}$ be the restriction of the parallel transport map along $\gamma$ to the orthogonal complement of $\gamma'(0)=\gamma'(1).$
$A$ is clearly an isometry and also it is orientation preserving. To see the latter just observe that when $e_1,...,e_{n-1}$ is a positively oriented ONB of the orthogonal complement of $\gamma'(0)$ and if $E_1(t),...,E_{n-1}(t)$ denote their parallel transports along $\gamma,$ then for all $t$ we have
\begin{equation}
\omega_{\gamma (t)}(E_1(t),...,E_{n-1}(t),\gamma '(t))\neq 0
\end{equation} because $E_1(t),...,E_{n-1}(t),\gamma'(t)$ is a basis for $T_{\gamma(t)}M$ and therefore, as $e_1,...,e_{n-1}$ is positively oriented,
\begin{equation}
\omega_{\gamma (t)}(E_1(t),...,E_{n-1}(t),\gamma '(t))>0.
\end{equation} So $E_1(1),...,E_{n-1}(1)$ has the same orientation as $e_1,...,e_{n-1}.$ Because of $E_i(1)=A(e_i),$ $A$ is orientation preserving.
Because $n$ is even, Lemma 3.8 in Do Carmo impliess that there is a $v\in T_{\gamma(0)}M$ such that
\begin{equation}
V(1)=A(v)=v,
\end{equation}where $V(t)$ denotes the parallel transport of $v$ along $\gamma.$
Now to the rest of the exercise:
Let $v$ and $V(t)$ such as we obtained it in 2).
Then
\begin{equation}
h(s,t)=exp_{\gamma(t)}(sV(t))
\end{equation} is a variation of $\gamma.$ By the same calculation as in the proof of the Synge-Weinstein-Theorem we get
\begin{equation}
\frac{1}{2}E''(0)=-\int_0^1K(V(t),\gamma'(t))dt<0
\end{equation}
and so there is a $s_0$ such that the closed curve $c(t):=h(s_0,t)$ satisfies $l(c)<l(\gamma)$ (details again at the end of the proof of the Synge-Weinstein-Theorem).
