In the context of next generation radio telescopes, like the Square Kilometre Array, the efficient processing of large-scale datasets is extremely important. Convex optimisation tasks under the compressive sensing framework have recently emerged and provide both enhanced image reconstruction quality and scalability to increasingly larger data sets. We focus herein mainly on scalability and propose two new convex optimisation algorithmic structures able to solve the convex optimisation tasks arising in radio-interferometric imaging. They rely on proximal splitting and forward-backward iterations and can be seen, by analogy with the CLEAN major-minor cycle, as running sophisticated CLEAN-like iterations in parallel in multiple data, prior, and image spaces. Both methods support any convex regularisation function, in particular the well studied ℓ1 priors promoting image sparsity in an adequate domain. Tailored for big-data, they employ parallel and distributed computations to achieve scalability, in terms of memory and computational requirements. One of them also exploits randomisation, over data blocks at each iteration, offering further flexibility. We present simulation results showing the feasibility of the proposed methods as well as their advantages compared to state-of-the-art algorithmic solvers. Our Matlab code is available online on GitHub.