We introduce MinConvLSTM and MinConvGRU, two novel spatiotemporal models that combine the spatial inductive biases of convolutional recurrent networks with the training efficiency of minimal, parallelizable RNNs. Our approach extends the log-domain prefix-sum formulation of MinLSTM and MinGRU to convolutional architectures, enabling fully parallel training while retaining localized spatial modeling. This eliminates the need for sequential hidden state updates during teacher forcing - a major bottleneck in conventional ConvRNN models. In addition, we incorporate an exponential gating mechanism inspired by the xLSTM architecture into the MinConvLSTM, which further simplifies the log-domain computation. Our models are structurally minimal and computationally efficient, with reduced parameter count and improved scalability. We evaluate our models on two spatiotemporal forecasting tasks: Navier-Stokes dynamics and real-world geopotential data. In terms of training speed, our architectures significantly outperform standard ConvLSTMs and ConvGRUs. Moreover, our models also achieve lower prediction errors in both domains, even in closed-loop autoregressive mode. These findings demonstrate that minimal recurrent structures, when combined with convolutional input aggregation, offer a compelling and efficient alternative for spatiotemporal sequence modeling, bridging the gap between recurrent simplicity and spatial complexity.