That you keep saying this does not make it true. I will run through your case with actual numbers. We must consider total taxes paid in the two cases: (A) parent converts, (B) parent does not convert.
Let's put the parent at the bottom of the 22% bracket and have the heir $50000 into the 32% bracket. (I chose those because think that makes your case as strong as possible, but I actually think it doesn't matter.) I am using 2025 tax tables.
Case A
Parent pays $11157+0.22*$10,000 = $13,157.
Heir pays =$80398+(50000+0)*0.32 = $96,398.
Total taxes paid: $109,755
Case B
Parent pays $11157+0.22*0 = $11,157.
Heir pays =$80398+(50000+10000)*0.32 = $99,598
Total taxes paid: $110,755
The only thing that changed was whether the parent converted or not. As a direct consequence of this one action or inaction, the difference in taxes paid was $110,755 - $109,755 = $1,000.
Hmm, why the even number? Because [$110,755 - $109,755]/$10,000 = 0.1 which also = 0.32 - 0.22.