Data series similarity search is a core operation for several data series analysis applications across many different domains. Nevertheless, even state-of-the-art techniques cannot provide the time performance required for large data series collections. We propose ParIS and ParIS+, the first disk-based data series indices carefully designed to inherently take advantage of multi-core architectures, in order to accelerate similarity search processing times. Our experiments demonstrate that ParIS+ completely removes the CPU latency during index construction for disk-resident data, and for exact query answering is up to 1 order of magnitude faster than the current state of the art index scan method, and up to 3 orders of magnitude faster than the optimized serial scan method. ParIS+ (which is an evolution of the ADS+ index) owes its efficiency to the effective use of multi-core and multi-socket architectures, in order to distribute and execute in parallel both index construction and query answering, and to the exploitation of the Single Instruction Multiple Data (SIMD) capabilities of modern CPUs, in order to further parallelize the execution of instructions inside each core.