We study low-temperature transport through carbon nanotube quantum dots in the Coulomb blockade regime coupled to niobium-based superconducting leads. We observe pronounced conductance peaks at finite source-drain bias, which we ascribe to elastic and inelastic cotunneling processes enhanced by the coherence peaks in the density of states of the superconducting leads. The inelastic cotunneling thresholds display a marked dependence on gate voltage caused by different tunneling renormalizations of the two subbands in the nanotube. Finally, we discuss the gate-dependent subgap structure observed in a strongly coupled device with odd electron occupation.