Reducing carbon emissions in the manufacturing sector is an important pathway to achieving sustainable development goals (SDGs). Anchored in SDG 13, this study focuses on the operational and low-carbon research and development (R&D) issues faced by firms in a supply chain, comprising one supplier and one manufacturer, in two scenarios: cooperative decision making and independent decision making. The independent decision-making scenario is classified into three modes: supplier-independent R&D, manufacturer-independent R&D, and two firms simultaneously R&D. First, this study evaluates the various modes with regard to unit abatement levels, corporate profits, environmental impact, and consumer surplus. Next, the analysis investigates the influence of carbon tax and R&D cost coefficient on equilibrium strategy. Our findings indicate that under the cooperative mode, when the cost coefficient is below a certain threshold, a high carbon tax can effectively incentivize both firms to increase their carbon reduction efforts, thereby enhancing the overall profitability of the supply chain and advancing the achievement of the SDG 13. The simultaneous independent R&D can enable all parties to obtain a high unit abatement level than when a single firm conducts independent R&D. In addition, the simultaneous R&D mode of two firms can generate higher profits and consumer surplus than the R&D mode of a single firm, and can bring higher total supply chain profits than the cooperative mode when the R&D cost is low. After extending modes to the case where the two firms cooperate only in the field of R&D, this study finds that the unit abatement level reaches the maximum.