Artificial reverberation (AR) models play a central role in various audio applications. Therefore, estimating the AR model parameters (ARPs) of a reference reverberation is a crucial task. Although a few recent deep-learning-based approaches have shown promising performance, their non-end-to-end training scheme prevents them from fully exploiting the potential of deep neural networks. This motivates the introduction of differentiable artificial reverberation (DAR) models, allowing loss gradients to be back-propagated end-to-end. However, implementing the AR models with their difference equations "as is" in the deep learning framework severely bottlenecks the training speed when executed with a parallel processor like GPU due to their infinite impulse response (IIR) components. We tackle this problem by replacing the IIR filters with finite impulse response (FIR) approximations with the frequency-sampling method. Using this technique, we implement three DAR models -- differentiable Filtered Velvet Noise (FVN), Advanced Filtered Velvet Noise (AFVN), and Delay Network (DN). For each AR model, we train its ARP estimation networks for analysis-synthesis (RIR-to-ARP) and blind estimation (reverberant-speech-to-ARP) task in an end-to-end manner with its DAR model counterpart. Experiment results show that the proposed method achieves consistent performance improvement over the non-end-to-end approaches in both objective metrics and subjective listening test results.