Q. But it also feels odd to include the same 1098T on both the parent's return and the student-dependent's return. So I want to confirm I have this right?
A. Yes, you have that right.
The 1098-T is only an informational document. The numbers on it are not required to be entered onto your tax return. However receipt of a 1098-T frequently means you are either eligible for a tuition credit or possibly your student has taxable scholarship income. So, sometimes the 1098-T goes on the parent's return, sometime it goes on the student's and sometime it goes on both.
You claim the tuition credit, or report scholarship income, based on your own financial records, not the 1098-T.
If you claim the tuition credit, you do need to report that you got one or that you qualify for an exception (the TurboTax interview will handle this)