I got interested in this question too, and this is how I understood it. I do not know the exact definition of a bundle over an orbicurve, but what you are asking should be the definition, and here is why.

Let $E$ be a bundle on a curve $X$. A *parabolic structure* at $x\in X$ is a sequence of bundles and maps

$E=E_0 \rightarrow E_1 \rightarrow \cdots \rightarrow E_{N-1} \rightarrow E_N=E(x)$

such that the composition is the canonical map $E\rightarrow E(x)$. If we drop the assumption that these are bundles and allow any coherent sheaves, but impose further that for any $i$ we have $E_i\to E_i(x)$ is the canonical map, we obtain the notion of a parabolic coherent sheaf.

Now let us take the direct sum

$\tilde{E}:=\bigoplus_{i=0}^{N-1} E_i$

If $t$ was the local parameter at $x$, we have the action of $t^{1/N}$ on $\tilde{E}$ as follows:

$t^{1/N}(s_1,\ldots,s_N) = (x s_N, s_1,\ldots,s_{N-1}).$

The group of $N$-th roots of unity $\mu_N$ acts on the choices of $t^{1/N}$, and also on $\tilde{E}$:

$\zeta(s_1,\ldots,s_N) = (s_1, \zeta s_2,\ldots,\zeta^{N-1} s_N).$

The action of $t^{1/N}$ should be the lifting of our bundle to the $N$-fold cover of a small disk around $x$, and the action of $\mu_N$ gives the equivariant structure. You can go back, i.e. given $\tilde{E}$, use the $\mu_N$ to decompose it into summands, and then multiplication by $t^{1/N}$ will give the maps. So parabolic structure = our bundle lifts to an equivariant bundle on the $N$-fold cover of a small disk around $x$.