Degradable polymers synthesized from biomasses have become increasingly appealing for their environmentally friendly and sustainable features. In this study, (R)-(−)-carvone, a natural monoterpene, and its hydrogenated derivative, (+)-dihydrocarvone, are oxidized to afford two epoxy compounds, termed as EK and MOO, respectively. The epoxides are subject to ring-opening copolymerization with cyclic anhydrides, including phthalic anhydride and cis-5-norbornene-exo-2,3-dicarboxylic anhydride, organocatalyzed by a phosphazene base. High or complete anhydride conversion can be reached at 100 °C when the epoxides are used in slight excesses. The resulting polyesters exhibit fully alternating sequence distribution, high glass transition temperature (108–142 °C), and two-step thermal degradation (above 200 °C) which is distinct from the polyesters derived from common epoxides. Compared with EK, MOO allows better controlled and distinctly higher molar mass (>10 kg mol−1), indicating much less profound (intramolecular) transesterification occurring on the polyester backbone. Most interestingly, the lactone moiety of MOO, generated together with the epoxide moiety by oxidation and previously shown polymerizable, stays intact during the copolymerization, which may constitute a simple strategy for utilizing bio-sourced bifunctional compounds chemoselectively and/or orthogonally.