  In this work, we manipulate the phase shift of a Josephson junction using a
parallel double quantum dot (QD). By employing a superconducting quantum
interference device, we determine how orbital hybridization and detuning affect
the current-phase relation in the Coulomb blockade regime. For weak
hybridization between the QDs, we find $\pi$ junction characteristics if at
least one QD has an unpaired electron. Notably the critical current is higher
when both QDs have an odd electron occupation. By increasing the inter-QD
hybridization the critical current is reduced, until eventually a $\pi$-0
transition occurs. A similar transition appears when detuning the QD levels at
finite hybridization. Based on a zero-bandwidth model, we argue that both cases
of phase-shift transitions can be understood considering an increased weight of
states with a double occupancy in the ground state and with the Cooper pair
transport dominated by local Andreev reflection.
