•Fitting to the Bloch manifold via dictionary matching is computationally intensive.•Deep networks can compactly do this via hierarchical partitioning and multiscale approximations.•We propose a 2-stage dictionary-matching-free pipeline for multi-parametric quantitative MRI.•Convex spatiotemporal reconstruction removes aliasing from compressed-sampled images.•A compact encoder-decoder model embeds the Bloch response model for quantitative inference. Display omitted We propose a dictionary-matching-free pipeline for multi-parametric quantitative MRI image computing. Our approach has two stages based on compressed sensing reconstruction and deep learned quantitative inference. The reconstruction phase is convex and incorporates efficient spatiotemporal regularisations within an accelerated iterative shrinkage algorithm. This minimises the under-sampling (aliasing) artefacts from aggressively short scan times. The learned quantitative inference phase is purely trained on physical simulations (Bloch equations) that are flexible for producing rich training samples. We propose a deep and compact encoder-decoder network with residual blocks in order to embed Bloch manifold projections through multi-scale piecewise affine approximations, and to replace the non-scalable dictionary-matching baseline. Tested on a number of datasets we demonstrate effectiveness of the proposed scheme for recovering accurate and consistent quantitative information from novel and aggressively subsampled 2D/3D quantitative MRI acquisition protocols.