been achieved, high-fidelity two-qubit gates in coupler architecture are still challenging. This paper pays special attention to examining the gate error sources and primarily concentrates on the related physical mechanism of ZZ parasitic couplings using a systematic effective Hamiltonian approach. Benefiting from the effective Hamiltonian, we provide simple and straightforward insight into the ZZ parasitic couplings that were investigated previously from numerical and experimental perspectives. The analytical results obtained provide exact quantitative conditions for eliminating ZZ parasitic couplings, and trigger four novel realizable parameter regions in which higher fidelity two-qubit gates are expected. Beyond the numerical simulation, we also successfully drive a simple analytical result of the two-qubit gate error from which the trade-off effect between qubit energy relaxation effects and ZZ parasitic couplings is understood, and the resulting two-qubit gate error can be estimated straightforwardly. Our study opens up new opportunities to implement high-fidelity two-qubit gates in superconducting coupler architecture.