Molecular simulations demonstrate that the enthalpic softening of elastomeric nanocomposites upon heating can arise naturally from a Poisson's ratio mismatch between elastomer and nanoparticle networks, providing a more parsimonious explanation for this phenomenon than the widely accepted interpretation based on glassy interparticle bridging. Despite a century of use, the mechanism of nanoparticle-driven mechanical reinforcement of elastomers is unresolved. A major hypothesis attributes it to glassy interparticle bridges, supported by an observed inversion of the variation of the modulus E(T) on heating - from entropic stiffening in elastomers to enthalpic softening in nanocomposites. Here, molecular simulations reveal that elastomer enthalpic softening can instead emerge from a competition over the preferred volumes between elastomer and nanoparticulate networks. A theory for this competition accounting for softening of the bulk modulus on heating predicts the simulated E(T) inversion, suggesting that reinforcement is driven by a volume-competition mechanism unique to cocontinuous systems of soft and rigid networks.