Operation strategies for a park-level integrated energy system (PIES) in terms of carbon prices and feed-in tariffs, have not been adequately studied. This paper addresses this knowledge gap by proposing operation strategies based on the PIES driven by biogas, solar energy, natural gas, and the power grid. Meanwhile, the electricity-driven dispatching strategy (EDS), thermal-driven dispatching strategy (TDS), cost-driven dispatching strategy (CDS) are compared to assess their impacts on operation cost, carbon dioxide emissions, etc. The flexibility and complementarity of the three operation strategies in energy supply are analyzed in detail. The results indicated that biogas was the main energy supply fuel, accounting for 46% to 72% of the total energy supply. About 33% to 54% of electricity was transmitted to the grid each month using the TDS. The annual initial capital cost of the CDS was only 1.39% higher than that of the EDS. However, the annual operation cost of the EDS was 16.86% higher than that of the CDS. The emissions of the EDS were the lowest, and the CDS had 38.51% higher emissions than the EDS. In the CDS, the ratio of carbon emission costs to operation costs was as high as 0.80 when the carbon tax reached USD 100/ton. The carbon tax had a greater impact on the CDS than the other strategies. Feed-in tariffs had a greater impact than the carbon tax on the TDS. This study provides an effective method for the selection of optimal operation strategies in regards to carbon prices and feed-in tariffs.