Controlling fermion parity in quantum-dot Josephson junctions (QDJJs) enables access to novel superconducting physics. While parity transitions are usually tuned electrically, we study phase-induced transitions in a strongly coupled QDJJ using the superconducting phase difference across the dot terminals, denoted by φ. We find that this phase allows control over the ground state parity, with parity transitions observed as the Andreev bound states (ABSs) energies crossing zero. These crossings occur near φ = π, and their behavior is in good correspondence to a simple model considering a single-level quantum dot proximitized by a superconductor. These results place phase-controlled QDJJ arrays as a promising platform for exploring exotic phenomena such as Majorana zero modes.