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Using non-Hermitian random matrices as a framework for description of universal characteristics of dissipative chaotic quantum many-body systems,  we address the problem of characterizing the parametric correlations of their spectral densities.  Considering parameter-dependent family of complex Ginibre matrices, we employ the method of exact replica integrability  (Kanzieper, 2002) to derive an explicit, closed-form expression for the parametric number variance in the systems of symmetry class A for eigenvalues in a circular domain containing on average a finite number of eigenvalues in the spectral bulk.  We also discuss a relation between parametric correlations of spectral densities and the distribution of the so-called eigenvector non-orthogonality factor, which attracted considerable interest in recent years. The talk will be based on a joint work with  Bertrand Lacroix-A-Chez-Toine.