Learn Farsi: مرد پیری که کراوات قرمز زده است، اخم کرده است. - The old man wearing a red tie is frowning.

This is a key feature of Farsi stative verbs. For conditions or postures, Farsi describes the present state by referring to the completed action that caused it. Literally, zade ast means "he has put on" (the tie), which Farsi uses to express the continuous, ongoing state of "he is wearing" it right now.

I know pushidan means "to wear." Why do we use a form of zadan for wearing a tie?

In Farsi, pushidan is strictly used for main articles of clothing that you physically put your body into, like shirts, pants, or jackets. For accessories that you attach, apply, or tie on—such as ties, glasses, watches, or perfume—you use zadan (literally "to hit" or "to apply").

Why is "is frowning" also in the Present Perfect tense (akhm karde ast) instead of a present continuous tense?

Just like wearing a tie, frowning is treated as a physical condition or posture. The man performed the action of frowning a moment ago, and he is currently still in that state. Therefore, Farsi uses the Present Perfect (literally "he has made a frown") to convey "he is currently frowning."

The English translation says "The old man", but mard-e piri has the indefinite -i suffix. Shouldn't that mean "An old man"?

Usually, yes. But here, the -i on piri is immediately followed by ke (who/that). This is known as the restrictive -i. Instead of meaning "some unknown old man," it singles out a specific one: "the [particular] old man who...". It signals to the listener that the exact identity of the man is about to be explained in the clause that follows.