  Parabolic SL(r,C)-opers were defined and investigated in [BDP] in the set-up
of vector bundles on curves with a parabolic structure over a divisor. Here we
introduce and study holomorphic differential operators between parabolic vector
bundles over curves. We consider the parabolic SL(r,C)-opers on a Riemann
surface X with given singular divisor S and with fixed parabolic weights
satisfying the condition that all parabolic weights at any point $x_i$ in S are
integral multiples of $\frac{1}{2N_i+1}$, where $N_i > 1$ are fixed integers.
We prove that this space of opers is canonically identified with the affine
space of holomorphic differential operators of order r between two natural
parabolic line bundles on X (depending only on the divisor S and the weights
$N_i$) satisfying the conditions that the principal symbol of the differential
operators is the constant function 1 and the sub-principal symbol vanishes
identically. The vanishing of the sub-principal symbol ensures that the
logarithmic connection on the rank r bundle is actually a logarithmic SL(r,
C)-connection.
