Last year, the average salary of a first-time (teacher was $29,564 annually, increasing 3.2 percent over the previous year).
B) teacher was $29,564 annually, an increase of 3.2 percent over that of
C) teacher was $29,564, an increase of 3.2 percent from

The answer is C. I understand that “annually” is redundant, but isn’t C an illogical comparison between salary and year? Are “that of” and “those of” only correct when used in an actual comparison (i.e. “like…” “-er”)? Thanks!

No, that’s an accepted form. Don’t let it drive you nuts–in general your instinct to want to make the comparison as parallel as possible is good! It’s just that this is not saying that $29,564 is more than last year, it’s saying the current salary is up from (where it was) last year.