  We study a Cooper-pair transistor realized by two Josephson weak links that
enclose a superconducting island in an InSb-Al hybrid nanowire. When the
nanowire is subject to a magnetic field, isolated subgap levels arise in the
superconducting island and, due to the Coulomb blockade,mediate a supercurrent
by coherent co-tunneling of Cooper pairs. We show that the supercurrent
resulting from such co-tunneling events exhibits, for low to moderate magnetic
fields, a phase offset that discriminates even and odd charge ground states on
the superconducting island. Notably,this phase offset persists when a subgap
state approaches zero energy and, based on theoretical considerations, permits
parity measurements of subgap states by supercurrent interferometry. Such
supercurrent parity measurements could, in a new series of experiments, provide
an alternative approach for manipulating and protecting quantum information
stored in the isolated subgap levels of superconducting islands.
