  We report the experimental manifestation of even-odd parity effects in the
transport characteristics of insulating Josephson junction chains which occur
as the superconducting gap is suppressed by applied magnetic fields at
millikelvin temperatures. The primary signature is a non-monotonic dependence
of the critical voltage, $V_c$, for the onset of charge transport through the
chain, with the parity crossover indicated by a maximum of $V_c$ at the parity
field $B^*$. We also observe a distinctive change in the transport
characteristics across the parity transition, indicative of Cooper-pair
dominated transport below $B^*$, giving way to single-electron dominated
transport above $B^*$. For fields applied in the plane of the superconducting
aluminum films, the parity effect is found to occur at the field, $B^*_{||}$,
such that the superconducting gap equals the single-electron charging energy,
$\Delta(B^*_{||})=E_C$. On the contrary, the parity effect for perpendicularly
applied fields can occur at relatively lower fields, $B^*_\perp\simeq
2\Phi_0/A_I$, depending only on island area, $A_I$. Our results suggest a novel
explanation for the insulating peak observed in disordered superconducting
films and one-dimensional strips patterned from such films.
