I'm looking to gather a few friends at the park to show a documentary on a projector screen. Is a permit required for that?

If the projector is portable, it does not require a permit; and if the gathering is limited (not large gathering) then a permit is not required; and if safe protocols are met (masks, 6 ft, etc.) then no permits are required.

Small gathering + portable projector with social distance No permit required

You should call the parks department and ask them

https://www.prospectpark.org/visit-the-park/general-info/permits/film-and-photo-shoots/

No as long as it's less than 20 people.

I'm not really sure about that