Everyone has made some good points that I was going to mention but to put it in terms that might make it easier to decide. As stated by others, paying off debt and being free is always the goal and desirable.
However, you must also consider the "efficiency" of what you do as well. For example, there are two common types of student loans (there are others but let's focus on these) and that is subsidized and unsubsidized.
The main difference? Subsidized loans don't earn interest on your balance while you are in school, it only happens when you graduate and come out of repayment grace period. Unsubsidized loans begin accumulating interest the moment they are disbursed, but you are not required to make payments on them until you graduate.
All student loans are deferred until you graduate and exhaust your grace period or other means of deferring your payment, say for example a postponement or forbearance. However, it is often recommended that on UNSUBSIDIZED loans, you pay down your principle while still in school to avoid that massive interest amount that will get added to it when you are officially in repayment. On the other hand, it is often (if not always) recommended that you hold off on paying SUBSIDIZED loans until you are done and go into repayment, as for all intents and purposes its not costing you anything extra to wait. Family and parent loans are considered and treated more like personal loans, so treat them as such.
Hope that helps. Also, don't forget to take advantage of the income based repayment options, as they will make the payments manageable enough to avoid making them a burden while you are trying to get a job and go post education.
Income-Driven Plans (Department of Education)
Income-Driven Repayment Plans (nelnet)