The wide deployment of phasor measurement units has enabled data-driven transient stability prediction (TSP) of power systems. However, ensuring the reliability of TSP results is still a significant challenge that limits the practical application of data-driven methods. To this end, a convolutional neural network (CNN)-based deep ensemble model with optimized class-specific thresholds is proposed to achieve reliable TSP. Specifically, a CNN is utilized as the backbone predictor, where the time-series variables from multiple generators are transformed into image-like inputs, and a CNN-based deep ensemble model is developed to provide accurate confidence estimation for TSP. Subsequently, considering the asymmetric importance of different classes in TSP, a confidence-based class-specific thresholds rule is adopted, and a multi-objective optimization model for determining the class-specific thresholds is formulated. In this optimization model, the reliability requirement of TSP is imposed as a constraint, requiring that true unstable rate (TUR) equal to 100%, with the objectives of minimizing the rejection rate and maximizing the true stable rate (TSR). The Pareto front of the class-specific thresholds can be obtained by solving the optimization model. Test results on two benchmark power systems show that the proposed method achieves a TUR of 100% and a TSR of at least 99% with approximately 10% of the samples rejected, demonstrating its effectiveness and scalability.