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Multicore and Parallel Systems

Parallel and multi-core processing both refer to the same thing: the ability to execute code at the same time (in more than one core/CPU/machine.) So in this sense multi-core is just a means to do parallel processing.

On the other hand, concurrency (which is probably what you mean by parallel processing) refers to having multiple units of execution (threads or processes) that are interleaved. This can also happen in either in a single core CPU or in many cores/CPUs or even in many machines (clusters).

Summing up, multicore is a subset of parallel and concurrency can occur with or without parallelism. The field that studies this is fistributed systems or distributed computing.