The Josephson quasiparticle (JQP) cycle in a voltage-biasedsuperconducting single-electron transistor (SSET) combinescoherent Cooper pair tunneling with incoherent quasiparticledecay. We have measured a striking modification of the JQPprocess in a system of two SSET's having a large mutualcapacitance. We find, among other effects, that the JQPcurrent in one SSET may be enhanced by the presence of aquasiparticle current through the other SSET. A simplified model of the coupled-SSET system is presented which reproducesthe enhancement effect.