Do I need to derate wire ampacity for two or more sets of wires in the same raceway for subpanel wiring?
Your final wire size selection for the ungrounded conductors was exactly right at #2 copper. Your neutrals, however, could have been much smaller. You are allowed to calculate the total unbalanced load and in turn size the neutral accordingly.
You still must apply the ambient temperature derating factor and you are required to at least match the size of the grounding conductor. For example, if your total calculated unbalanced load on the neutral was 25 amps and the ambient temperature was negligible, the ampacity table would allow for only a #10 AWG wire to carryn the neutral current, but because the neutral cannot be smaller than the grounding conductor you would still be required to use a #6 AWG. I have the tables underlined in my codebook but don't have it on hand , sorry. 
 The minimum size grounding conductor chart chart is in article 250. 
Just found this on Mike Holt's website: { The grounded (neutral) service conductor must be sized to carry the maximum unbalanced load in accordance with 220.22 and must not be sized smaller than required by 250.24(B).}