High-precision electrostatic field calculations are crucial to the design and reliability assessment of insulation structures in power equipment. A posteriori error estimation can effectively quantify the accuracy of finite element numerical solutions. However, research on a posteriori error estimation specifically targeting typical electrostatic problems in power equipment remains relatively limited. In this paper, the explicit residual-based error estimator (RBE) and the superconvergent patch recovery-based Zienkiewicz-Zhu estimator (SPR-ZZ) are constructed using a two-dimensional parallel-plate capacitor as a typical benchmark example. The results show that RBE and SPR-ZZ can serve as effective a posteriori error estimation techniques in finite element analysis of electrostatic fields, capable of accurately identifying high-error regions. RBE is highly sensitive to geometric singularities or regions with abrupt changes in the physical field, enabling rapid identification and marking of potential high-error zones. In contrast, SPR-ZZ delivers accurate and smooth error estimates. It is better suited for finely capturing regions with intense electric field variations but no obvious geometric singularities. RBE and SPR-ZZ offer a more efficient alternative to traditional grid convergence studies by enabling error-driven adaptive refinement, significantly reducing computational cost while ensuring accuracy.