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Let $X$ be a compact Riemann surface of genus $g \geq 3$ and $S$ a finitesubset of $X$. Let $\xi$ be fixed a holomorphic line bundle over $X$ of degree$d$. Let $\mathcal{M}_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}_{pc}(r,\alpha, \xi)$ ) denote the moduli space of parabolic connections of rank $r$,degree $d$ and full flag rational generic weight system $\alpha$,(respectively, with the fixed determinant $\xi$) singular over the parabolicpoints $S \subset X$.  Let $\mathcal{M}'_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}'_{pc}(r,\alpha, \xi)$) be the Zariski dense open subset of $\mathcal{M}_{pc}(r, d,\alpha)$ (respectively, $\mathcal{M}_{pc}(r, \alpha, \xi)$ )parametrizing allparabolic connections such that the underlying parabolic bundle is stable. Weshow that there is a natural compactification of the moduli spaces$\mathcal{M}'_{pc}(r, d, \alpha)$, and $\mathcal{M}'_{pc}(r, \alpha, \xi)$ bysmooth divisors.  We describe the numerically effectiveness of these divisors at infinity. Wedetermine the Picard group of the moduli spaces $\mathcal{M}_{pc}(r, d,\alpha)$, and $\mathcal{M}_{pc}(r, \alpha, \xi)$. Let $\mathcal{C}(L)$ denotethe space of holomorphic connections on an ample line bundle $L$ over themoduli space $\mathcal{M}(r, d, \alpha)$ of parabolic bundles. We show that$\mathcal{C}(L)$ does not admit any non-constant algebraic function.

Line bundles on the moduli space of parabolic connections over a compact Riemann surface