Parallel-wound no-insulation (PW-NI) high-temperature superconducting (HTS) coils significantly reduce charging delay while maintaining excellent self-protection capability, demonstrating great potential for high-field applications. Existing models that couple the T-A formulation with equivalent circuits have demonstrated high accuracy in electromagnetic analysis of PW-NI coils. However, eliminating the computational overhead caused by frequent variable mapping and data exchange between electromagnetic and circuit modules is important for improving computational efficiency, particularly in long-duration transient simulations of large-scale magnets. To address this issue, an extended T-A formulation based on potential-chain recursion, termed PCR-TA, is proposed. By directly embedding inter-tape current sharing and radial current bypass behaviors into the finite-element framework, this method computes the transient electromagnetic response of PW-NI coils without requiring an explicit equivalent circuit model. Building upon it, a multi-scale approach is further developed for large-scale PW-NI coils. The validity of the proposed method and its multi-scale extension is verified through comparisons with experimental measurements and field-circuit coupled modeling results. Comparative analyses demonstrate that the PCR-TA method achieves a speedup of approximately 2.4 over the field-circuit coupled method, whereas its multi-scale extension further increases this speedup to roughly 5.8. Furthermore, the PCR-TA method is extended to model the continuous transition of PW-NI coils from power-supply charging to closed-loop operation. This work provides an efficient method and tool for the electromagnetic modeling of PW-NI coils under both driven and closed-loop operating conditions.