This paper investigates a symbiotic unmanned aerial vehicle (UAV)-assisted intelligent reflecting surface (IRS) radio system, where the UAV is leveraged to help the IRS reflect its own signals to the base station, and meanwhile enhance the UAV transmission by passive beamforming at the IRS. First, we consider the weighted sum bit error rate (BER) minimization problem among all IRSs by jointly optimizing the UAV trajectory, IRS phase shift matrix, and IRS scheduling, subject to the minimum primary rate requirements. To tackle this complicated problem, a relaxation-based algorithm is proposed. We prove that the converged relaxation scheduling variables are binary, which means that no reconstruct strategy is needed, and thus the UAV rate constraints are automatically satisfied. Second, we consider the fairness BER optimization problem. We find that the 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 designs under different setups, as compared with benchmarks.