2 added 106 characters in body
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This seemed to work for me.

ParallelEvaluate[Unprotect@Print];
ParallelEvaluate[Print = Null &];
ParallelDo[Print[{i, $KernelID}], {i, 1, 10}]

This also seemed to work:

ParallelDo[With[{Print = Null &}, Print[{i, $KernelID}]], {i, 1, 10}]

This seemed to work for me.

ParallelEvaluate[Unprotect@Print];
ParallelEvaluate[Print = Null &];
ParallelDo[Print[{i, $KernelID}], {i, 1, 10}]

This seemed to work for me.

ParallelEvaluate[Unprotect@Print];
ParallelEvaluate[Print = Null &];
ParallelDo[Print[{i, $KernelID}], {i, 1, 10}]

This also seemed to work:

ParallelDo[With[{Print = Null &}, Print[{i, $KernelID}]], {i, 1, 10}]
1
source | link

This seemed to work for me.

ParallelEvaluate[Unprotect@Print];
ParallelEvaluate[Print = Null &];
ParallelDo[Print[{i, $KernelID}], {i, 1, 10}]