The moduli space ${\frak N}^{c}_{g,1}$ of non-orientable surfaces of genus $g \geq 0$ with $c \geq 0$ distinguished points and one boundary curve is described via a model ${\frak P}(h,c)$: the space of configurations of $h=g+c+1$ pairs of parallel slits in $\mathbb{C}$. Based on this model, we prove that ${\frak N}^{c}_{g,1}$ is a non-orientable manifold, and we compute its homology for $h \leq 3$ with ${\mathbb Z}_2$-coefficients.