<p>I have a Printrboard rev. D which includes Allegro A4982 drivers. I would like to replace them, but it appears that newer TMC drivers all require several pins for proper operation.</p>

<p><a href="https://i.stack.imgur.com/ZnRf5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZnRf5.png" alt="enter image description here"></a></p>

<p>How can I upgrade the drivers in the Printrboard rev. D?</p>

You can't. In fact, pretty much all boards with directly soldered drivers are not upgradable.
I'd recommend to buy a proper, contemporary, customizable board like the SKR 1.3 for TMCs.
The Fysetc F6 would also work, but uses "custom" connectors and is still AVR based.