LaunchKernels[16];

ParallelEvaluate[

n[i_] = {};

M[i_] = {};

Do[
   AppendTo[n[i],{0,0}];
   AppendTo[M[i],{0,0}],
   {i,1,5}
  ]

];

Do[
   ParallelDo[
    J1[i] = {i, j};
    J2[i] = {i + 1, j + 1};
    AppendTo[M[i], J1[i]];
    AppendTo[n[i], J2[i]],
    {i, 1, 5}
   ],
  {j, 0, 5}
]

Join@@ParallelEvaluate[n[1]] = {{0, 0}, {0, 0}, {0, 0}, {0, 0},
                               {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0},
                               {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0},
                               {0, 0}, {0, 0}, {2, 1}, {2, 2}, {2, 3},
                               {2, 4}, {2, 5}, {2, 6}}