The section that you are referring to, Internal Revenue Code §3121(b)(10), allows student workers to not pay FICA tax on their wages. This section was originally set up to assist those who receive financial aid in the form of work-study.
Here is a good description of it from the IRS: https://www.irs.gov/charities-non-profits/student-exception-to-fica-tax
Here is the actual regulation: https://www.law.cornell.edu/cfr/text/26/31.3121%28b%29%2810%29-2
In your case, this may not apply. To qualify, your employment has to be incident to your studies, in otherwords, minor. Full-time, or significant part time work are NOT considered incident, according to the law and the cases thereunder.
Additionally, if you are teaching as a professor or adjunct faculty or teacher's aide, you would be considered a professional employee, which is by definition not an exempt study.
So based on two criteria, your university is correct. As I stated earlier, it is usually for undergraduate work study as part of a financial aid package.
I hope this has answered your question. If you have any more, please feel free to ask and I will be happy to answer.
Thanks! Have a great week!