I recently created a parallel implementation of the Merge Sort, in which the sorting of several groups was accomplished by different processes, and was wondering if this was theoretically possible on Turing Machine?

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This is a subtle question.

TMs are very much a sequential model of computation. So in some sense, TMs cannot (directly) model multiprocessing. However, TMs can do step-by-step simulations of the reductions a multiprocessor is carrying out. So TMs can do a sequential simulation of a parallel computation. Whether this a genuine model of parallel computation is debatable.

See also the discussion Applicability of Church-Turing thesis to interactive models of computation.

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