As far as I can tell, parent's use of "effective tax rate" is correct (the tuition waiver is not income in any meaningful sense -- especially for phd students who aren't even taking courses). So I'm not really sure what your point is?
I'm neither the OP nor do I know about his particular situation, but assuming a fairly generous stipend of $30,000
- Standard deduction is $6,350
- Personal exemption $4,050 x 3 (self + 2 kids) = $12,150
- Taxable income is $11,500
- Of that, it's 10% on the first $9,325 + 15% on the rest (up to $37,950), for a total of $1259. That's about 5%, not including state taxes.
Now, add in $40,000 of "benefits", most of which are not actually beneficial. Taxable income is now $51,500 and the total federal tax is about $8,614, which is a pretty substantial chunk of the 30,000 actual dollars he gets.
To make matters worse, a taxable income of $51,500 would make OP ineligible for some credits/deductions. For example, it's well past the EITC phase-out.