No, there is no 13th grade. The final grade of high school is 12th grade and any school that the student attends after that is not considered 13th grade. School after high school is considered college and completely optional.
Oddly enough the college years also goes by the same names as the high school grade names.
The first year of college is also freshman year, followed by sophomore year, junior year, and finally senior year.
I explain high school a little more in depth in this article I wrote.
However, the college grade levels that are optional after high school (12th grade) are never referred to as 13th, 14th, 15th, or 16th grade.