This paper investigates an unmanned aerial vehicle (UAV)-assisted symbiotic radio system with intelligent reflecting surface (IRS), where the UAV is leveraged to help the IRS forward its own signals to the base station, and meanwhile enhance primary network. By considering the fairness among IRSs, we aim to minimize the maximization of bit error rate (BER) by jointly optimizing the UAV trajectory, IRS phase shifts, and IRS scheduling, subject to the minimum primary rate requirements. The resultant problem is a mixed integer and non-convex optimization problem. We find that the commonly used relaxation-based method cannot solve this fairness BER problem since the minimum primary rate requirements may not be satisfied by the binary reconstruction operation. To address this issue, we first transform the binary constraints into a series of equivalent equality constraints. Then, a penalty-based algorithm is proposed to obtain a suboptimal solution. Numerical results are provided to evaluate the performance of the proposed design under different setups, as compared with benchmark schemes.