Like many entertainers, members of an advocacy association of musicians said that they no longer wanted to be tax evaders but instead wanted to begin paying into Social Security and building good credit histories.
A. Like many entertainers, members of an advocacy association of musicians said that they no longer wanted to be tax evaders-> comparison between "entertainers" and "members". It is correct. They refers to entertainers, which makes sense too. Let's keep it.
B. Like many other entertainers, tax evaders and members of an advocacy association of musicians said that they no longer wanted to be that -> comparison between "entertainers" and "tax evaders and members". Incorrect.
C. Members of an advocacy association of musicians said that they no longer wanted to be tax evaders like many entertainers -> comparison between "tax evaders" and "entertainers". Incorrect.
D. Members of an advocacy association of musicians said that they no longer wanted to be like tax evaders and many other entertainers -> comparison between "tax evaders" and "other entertainers". Incorrect.
E. Like many other entertainers, members of an advocacy association of musicians said that they no longer wanted to be tax evaders -> comparison between "other entertainers" and "members". It is better.
Now, if we compare A and E both are correct, but intended meaning is better in E.
So, I think E.