Let $E \to M$ be a vector bundle with connection $\nabla$, $\gamma \colon [0,1] \to M$ a smooth curve. A section $s = \gamma^* \tilde s$, $\tilde s \in \Gamma(E)$ of $\gamma^* E$ is called parallel if $$ \dot a^j(x(t)) + \Gamma^j_{ik}(x(t)) a^i(x(t))\dot x^i(t) = 0, \quad j = 1,\ldots,n, \quad t \in (0,1). \tag{1} $$ where $\gamma(t) = (x^i(t))$ in local coordinates on $M$, $\tilde s = (x^i,a^j(x))$ in a trivialising neighborhood, $\Gamma^j_{ik}$ are Christoffel symbols for $\nabla$.
We solve the equation (1) for the function $a^j(t) := a^j(x(t))$ with some initial condition. But it seems that it is possible that for two different $t_1$ and $t_2$ with $x(t_1) = x(t_2)$ we will obtain $a^j(t_1) \neq a^j(t_2)$. How to deal with such situations?
Probably, you're referring to the uniqueness of solutions of ordinary differential equations, so an orbit of the flow associated with the o.d.e. can't cross itself.
But I don't think that's a problem here, because the section $s(t) = (x(t), a(t))$ consists of the base point $x(t)$ and the tangent vector $a(t)$. So if, for $t_1 \neq t_2$, we have $x(t_1) = x(t_2)$ and $a(t_1) \neq a(t_2)$, we still have $s(t_1) \neq s(t_2)$, which is perfectly valid.
If we have $x(t_1) = x(t_2)$ and $a(t_1) = a(t_2)$, i.e. $s(t_1) = s(t_2)$, then the orbit is a closed loop.
In fact, the situation $x(t_1) = x(t_2)$ for $t_1 \neq t_2$ is exactly the concept of holonomy (http://en.wikipedia.org/wiki/Holonomy) where parallel transport along closed loops is considered.