According to historians, Charles Dickens was perhaps the first person to become a celebrity
not only for his theatrical performances as for his written works.
A) not only for his theatrical performances as for his written works -> not only X but also Y is correct form. It is not used here. Incorrect.
B) as much for his theatrical performances as did his written works -> as much x as y, so x and y should be parallel. In this option, "for his theatrical performances" is not parallel to "did his written works". Incorrect.
C) as much for his theatrical performances as for his written works -> Parallelism is corrected. Let's keep it.
D) equally for his theatrical performances as for his written works -> equally x as y is not correct form. Incorrect.
E) not so much for his theatrical performances as did his written works -> Again, we have parallelism error. Incorrect.
So, I think C.