We study the properties of bisqueezed tripartite Gaussian states created by two spontaneous parametric down-conversion processes that share a common idler. We give a complete description of the quantum correlations across all partitions, as well as of the genuine multipartite entanglement, obtaining analytical expressions for most of the quantities of interest. We find that the state contains genuine tripartite entanglement, in addition to the bipartite entanglement among the modes that are directly squeezed. We also investigate the effect of homodyne detection of the photons in the common idler mode, and analyze the final reduced state of the remaining two signal modes. We find that this measurement leads to a conversion of the coherence of the two signal modes into entanglement, a phenomenon that can be regarded as a redistribution of quantum resources between the modes. The applications of these results to quantum optics and circuit quantum electrodynamics platforms are also discussed.