Learned wavelet image and video coding approaches provide an explainable framework with a latent space corresponding to a wavelet decomposition. The wavelet image coder iWave++ achieves state-of-the-art performance and has been employed for various compression tasks, including lossy as well as lossless image, video, and medical data compression. However, the approaches suffer from slow decoding speed due to the autoregressive context model used in iWave++. In this paper, we show how a parallelized context model can be integrated into the iWave++ framework. Our experimental results demonstrate a speedup factor of over 350 and 240 for image and video compression, respectively. At the same time, the rate-distortion performance in terms of Bj{\o}ntegaard delta bitrate is slightly worse by 1.5\% for image coding and 1\% for video coding. In addition, we analyze the learned wavelet decomposition by visualizing its subband impulse responses.