table = ParallelTable[
  Quiet[
   {{q, t},
    1/Sqrt[2 Pi] NIntegrate[Exp[-I w t] f[q, w] /. {d1 -> 0.1, d2 -> 0.2, gamma1 -> 0.1, gamma2 -> 0.2, R -> 0.7},
      {w, -Infinity, Infinity},
      MaxRecursion -> 25]}
   ]
  ,
  {q, -qgrid, qgrid, res}, {t, 0, tgrid, res}
  ]

finterpolation = Interpolation[Flatten[table, 1], InterpolationOrder -> 3];