With the global climate change, the significance of carbon emission reduction is widely recognized. As an effective tool for emission reduction, carbon trading needs to be considered in the operation of the integrated power and transportation system (IPTS). This paper proposes a distributed risk-averse optimal dispatch scheme for IPTS considering carbon trading. First, a carbon trading framework for IPTS is developed, in which the microgrids (MGs) within the power distribution system (PDS) and the electric vehicle agent (EVA) of the transportation system (TS) jointly participate in the carbon trading market. Furthermore, the conditional value-at-risk (CVaR) method is used in the proposed scheme to manage the risks arising from diverse uncertainties, including power load, renewable power generation, traffic demand, and carbon price. To protect the privacy of the PDS and TS and improve computational performance, the spectral penalty parameter based alternating direction method of multipliers (S-ADMM) distributed solution method is developed to solve the IPTS dispatch model. Numerical results on two test systems illustrate the economic and carbon reduction benefits, risk management ability, and computational performance of the proposed scheme.