To meet the challenge of global climate change, the low-carbon transformation of China's industrial parks is imperative. Driven by the carbon peaking and carbon neutrality goals, distributed photovoltaic, wind power, and other renewable energies are connected to the industrial park on a large scale. The intermittent, fluctuating, and uncertain outputs have induced the surging demand for energy storage in the low-carbon park. The industrial park faces the dilemma between expansive investment in energy storage and the limitation of carbon emission. Therefore, this paper introduces the super-quantile to measure the carbon emission risk of industrial parks with uncertainties, proposes the carbon emission risk index, transforms its integral form into the linear constraint form that can be embedded in the energy storage planning model, and then establishes the energy storage capacity planning model of low-carbon parks considering the carbon emission risk constraint. Through the analysis of an industrial park, the effectiveness of the proposed energy storage capacity planning method is verified, and the carbon emission risk of the industrial park is reduced under the condition of a slight increase in the annual comprehensive cost. Finally, the sensitivity of risk threshold and confidence level is analyzed.