We construct the moduli spaces $M(n)$ of semistable parabolic sheaves of rank $n$ and fixed Euler characteristic on a disjoint union $X$ of integral projective curves with parabolic structures over Cartier divisors on $X$. In the case where $X$ is non-singular, $M$ is a normal projective variety. Suppose that $X$ is the desingularisation of a reducible reduced curve $Y$ with at most ordinary double points as singularities. We show that, for a suitable choice of parabolic structure, $M(n)$ is the normalisation of the moduli space of torsion-free sheaves of rank $n$ and fixed Euler characteristic on $Y$, and it is a desingularisation if semistability coincides with stability. We find explicit descriptions of $M(n)$ for small $n$ in some cases.