In this paper, we explored the estimation of the stress–strength reliability parameter R = P(Y < X) when the stress and strength variables were independent and followed the Discrete Alpha-Power Weibull (DAPW) distribution. Statistical inference was developed under three practically important incomplete data structures: Type-Ⅱ censored samples, upper record values, and mixed upper–lower record data. Classical and Bayesian estimation frameworks were employed. Maximum likelihood estimators (MLE) were obtained via numerical optimization, while Bayesian estimators were derived using non-informative prior distributions under the squared error loss function. To evaluate the finite-sample performance of the proposed estimators, a large-scale Monte Carlo simulation study was conducted, comparing bias, MSE, and interval precision across sample sizes and data structures. Bootstrap confidence intervals were constructed to quantify estimation uncertainty. The simulation results indicated that, within the considered settings, inference based on upper record values generally exhibited improved performance compared with Type-Ⅱ censored samples and, in many configurations, also compared with the mixed record scheme, in terms of bias, MSE, and interval length. Moreover, Bayesian estimators tended to demonstrate superior performance relative to their classical counterparts, particularly for small and moderate sample sizes. The practical applicability of the proposed methodology was illustrated through the analysis of a real data set on breakdown voltages of high-voltage capacitors, where the Bayesian approach based on upper record values provided competitive and precise estimates of the stress–strength reliability parameter.