Boring a Steep Parabola Into Bronze Round Stock with Lathe

Hello,

I am looking to drill/bore a hole about an inch deep into brass round stock with a lathe. My problem is, I cant find the tool I need because I don't know what it's called. Are there drill bits with a parabolic tip? I found a "tree shaped burr," which is the basic shape I need, but I don't think it is for drilling. Basically, the hole needs to be parallel at the top, and parabolic at the bottom. The general shape I am looking for is in the attachment photo.

Basically, the hole should look like an inverted pontiff's hat.

Any help would be appreciated.

Thanks,

Jesse