Nonreciprocal supercurrents in interacting nanostructures can arise when the positive and negative critical currents probe inequivalent portions of a phase-dependent many-body spectrum. We study this mechanism in a parallel double-quantum-dot Josephson junction with strong intradot Coulomb repulsion, where local and non-local Cooper-pair transfer coexist. Using exact diagonalization of a zero-bandwidth Hamiltonian with explicit superconducting orbitals, we show that orbital flux and the gauge-invariant tunnel-sign parity control the interference between local Josephson processes and non-local exchange. For detuned dots, this produces flux-tunable singlet, triplet, and doublet ground-state spectral branches and two rectification regimes. Near the boundaries of the charge sector with singly occupied dots [(1,1) sector], doublet branches compete with singlet or triplet branches, yielding diode efficiencies approaching 40%. Within the triplet-dominated (1,1) sector, nearby singlet crossings asymmetrically reshape the ground-state envelope even when both critical-current extrema lie on the triplet-like branch, producing a broader response of order 10-15%. Finite temperature generally suppresses charge-crossover rectification, however, for intermediate temperatures we find an enhancement of both the normalized efficiency and the absolute critical-current asymmetry in the triplet regime. These results establish phase-dependent many-body branch competition as a gate-, flux-, and tunnel-parity-sensitive source of Josephson nonreciprocity.