  We have made a systematic investigation of charge transport in 1D chains of
Josephson junctions where the characteristic Josephson energy is much less than
the single-island Cooper-pair charging energy, $E_\mathrm{J}\ll E_{CP}$. Such
chains are deep in the insulating state, where superconducting phase coherence
across the chain is absent, and a voltage threshold for conduction is observed
at the lowest temperatures. We find that Cooper-pair tunneling in such chains
is completely suppressed. Instead, charge transport is dominated by tunneling
of single electrons, which is very sensitive to the presence of BCS
quasiparticles on the superconducting islands of the chain. Consequently we
observe a strong parity effect, where the threshold voltage vanishes sharply at
a characteristic parity temperature $T^*$, which is significantly lower than
the the critical temperature, $T_c$. A measurable and thermally-activated
zero-bias conductance appears above $T^*$, with an activation energy equal to
the superconducting gap, confirming the role of thermally-excited
quasiparticles. Conduction below $T^*$ and above the voltage threshold occurs
via injection of single electrons/holes into the Cooper-pair insulator, forming
a non-equilibrium steady state with a significantly enhanced effective
temperature. Our results explicitly show that single-electron transport
dominates deep in the insulating state of Josephson-junction arrays. This
conduction process has mostly been ignored in previous studies of both
superconducting junction arrays and granular superconducting films below the
superconductor-insulator quantum phase transition.
