Does a man's driving his unveiled sister to her work—where she mixes with men—constitute choosing the lesser of two evils (the evil of mixing and unveiling at work instead of the two evils of unemployment and being alone with a driver), or is it cooperation in sin, especially given the parents' anger if he does not drive her?

If your refusal to drive your sister prevents her from displaying her adornments (tabarruj), then it is not permissible for you to drive her, because that prevents evil. However, if she will go out displaying her adornments in any case, then you must drive her, and it may become obligatory if your parents command you, to protect her from unlawful seclusion (khalwa) and harassment, while continuing to advise her gently and wisely, so that she may be guided.