Tree-based demappers for multiple-input multiple-output (MIMO) detection such as the sphere decoder can achieve near-optimal performance but incur high computational cost due to their sequential nature. In this paper, we propose the perturbed linear demapper (PLM), which is a novel data-driven model for computing soft outputs in parallel. To achieve this, the PLM learns a distribution centered on an initial linear estimate and a log-likelihood ratio clipping parameter using end-to-end Bayesian optimization. Furthermore, we show that lattice-reduction can be naturally incorporated into the PLM pipeline, which allows to trade off computational cost against coded block error rate reduction. We find that the optimized PLM can achieve near maximum-likelihood (ML) performance in Rayleigh channels, making it an efficient alternative to tree-based demappers.