Assuming that "console" means the Messages notebook window (and not, say, stdout when run in a terminal session), try something like the following
SetOptions[$FrontEnd, MessageOptions -> {"KernelMessageAction" -> "PrintToConsole"}]
$KernelCount; (* load the Parallel` code *)
Unprotect[MessagePacket]; (* patch MessagePacket upvalue *)
MessagePacket /:
Parallel`Protected`PacketHandler[
t : MessagePacket[sym_, tag_, text_], Parallel`Kernels`kernel_] :=
NotebookWrite[MessagesNotebook[],
Cell[text, "Message", "MSG",
CellLabel -> Parallel`Kernels`Private`kernelString[Parallel`Kernels`kernel],
ShowCellLabel -> True]]
ParallelEvaluate[1/0]
(* {ComplexInfinity, ComplexInfinity, ComplexInfinity, ComplexInfinity} *)