Comprehending the principles underlying the interaction between DNA molecules in electrolyte solution is of crucial importance for understanding phenomena of DNA condensation, recognition of homologous genes, and properties of DNA liquid crystals. Although forces acting between DNA molecules must be affected by the structure of the liquid separating them, even after DNA was considered in all its helical complexity over the last two decades, only until recently a primitive model of aqueous solution was used. In this paper we make a step towards unravelling earlier neglected effects of water structure on DNA-DNA interaction. We develop a model accounting for the nonlocal dielectric response of water and of the electrolyte plasma dissolved in it. We focus particularly on the study of the effect of dipolar overscreening, which leads to decaying oscillations of the electrostatic potential about DNA, and how it will affect electrostatic energy surfaces as a function of interaxial separation and mutual azimuthal orientation of the parallelly aligned interacting DNA. We found that going beyond the classical electrostatic description of water (the so-called primitive model) reveals a complex oscillatory interaction surface in space, with many possible metastable configurations. These however rapidly vanish with smearing of the DNA charge distribution, due to dephasing of the oscillatory components of the interaction. This dephasing results in the suppression of oscillation amplitudes under the dominance of non-oscillatory components in spatial dipolar correlations and the Debye screening. The dominant correlation patterns lead to a substantial enhancement of the difference between the interaction of homologous and nonhomologous DNA sequences, deepening and sharpening the homology recognition well.