I would not call them sufficient conditions but they are not necessary assumptions. The argument basically tells us that because A or B cannot do the assignment, C must do the assignment. A logical question is why cannot someone else do the assignment? We must assume that these are the only people that could do the assignment. In the absence of this assumption, the conclusion that C (Parker) must do the assignment no longer makes sense. (C) and (D) appear to support the premises - that neither L nor F can do the assignment, but these assumptions have no impact on the conclusion -that P has to be the one to do the assignment.