Can a parent claim a private student loan when the parent is the co-borrower as payment using a 1098-T?

You can deduct the student loan interest of a loan you co-signed if the student is your dependent or was your dependent when you signed for the loan AND you are not a dependent now or filing a married-separate return.

As the parent of a student-dependent,  you may "claim the 1098-T", even if the tuition was paid by student loans, regardless of whether the loans are only in the students name or if you co-signed. 

That is, as long as the student is your dependent, you may claim the tuition credit, even if the student was the one who paid the tuition. 

If you asking about claiming the student loan interest deduction, then it is required that you be a co-borrower to do so.