Is it? https://en.wikipedia.org/wiki/Persian_alphabet
Different examples given there use both variants.

The simplest answer is that this is just different, like a different choice of fonts. Compare American "z" and European "ƶ" in handwriting (or 7 vs. 7̵also).

While most extra marks in the Arabic script serve some purpose (for example, distinguishing B vs. T, etc.) that particular mark on K doesn't really have any purpose, so why not remove it?

Asking "why" in Linguistics does not usually lead to informative answers. "Because it happened" is the only real answer. Just different traditions.

The only guess I would have here is that maybe to avoid confusion (or increase parallelism) with the additional letter گ (https://en.wiktionary.org/wiki/%DA%AF) the extra shape on K was removed.

That makes a lot of sense. But do you know if the timeline of adopting the Arabic script in Persia matches up with the changes in kāf in Arabic?

I'm not sure either, but let's see--

Wikipedia says the Arabic script was adopted around the 9th century (so ~1200 years ago)
https://en.wikipedia.org/wiki/Persian_alphabet

So the question is just whether the extra symbol was added in Arabic before or after that point--

This page shows the history and variation but doesn't give a timeline:
https://en.wikipedia.org/wiki/Kaph

Based on a quick search, I'm not finding information about the date when the Arabic letter changed. The adoption of the script in Persia is nearly as old as the Qur'an itself though, so it's entirely possible that you're right about this, panini, assuming the earlier form was used consistently in the Qur'an.

That makes sense then! Interesting.