Learn Hardcore Finnish: Pysyn puistossa, kunnes aurinko laskee.

Finnish usually doesn’t need subject pronouns, because the verb ending already shows the person.
So pysyn already implies minä. You can add minä for emphasis or contrast:

Minä pysyn puistossa... = I (specifically) will stay...

Why is it puistossa and not puistossaN / puistoon / puistoa?

puistossa is inessive case (-ssa/-ssä) meaning in the park (location inside).
Other common options would change the meaning:

puistoon (illative) = into the park (movement toward/into)
puistossa (inessive) = in the park (staying there)
puistosta (elative) = out of the park
puistoa (partitive) wouldn’t fit the meaning in the park.

Why is the ending -ssa (puistossa) and not -ssä?

This is vowel harmony. Words with back vowels (a, o, u) take -ssa, while words with only front vowels (ä, ö, y) take -ssä.
puisto contains o (a back vowel), so it takes -ssa → puistossa.

What does kunnes do, and how is it different from kun?

kunnes means until and sets an endpoint for the action in the main clause:

Pysyn puistossa, kunnes aurinko laskee. = I stay there up to the moment when the sun sets.

kun usually means when (time) or sometimes because depending on context:

Kun aurinko laskee, lähden kotiin. = When the sun sets, I leave home.

So kunnes is specifically until, not just when.

Why is there a comma before kunnes?

Finnish normally uses a comma to separate a main clause and a subordinate clause:

Main clause: Pysyn puistossa
Subordinate clause: kunnes aurinko laskee

So the comma is standard here.

Why is aurinko laskee in the present tense if the meaning is about the future?

Finnish often uses the present tense to talk about the future when the time is clear from context (especially in time clauses like until/when):

aurinko laskee literally “the sun sets,” but it can mean “the sun will set.”

No special future tense is required.

What is laskee exactly—what verb is it from, and what form is it?

laskee comes from the verb laskea. In this context laskea means to set (about the sun).
laskee is 3rd person singular present:

aurinko laskee = the sun sets / will set.

Could I switch the clause order: Kunnes aurinko laskee, pysyn puistossa?

Yes, that’s grammatical. The meaning is essentially the same.
When the subordinate clause comes first, the comma still stays:

Kunnes aurinko laskee, pysyn puistossa.

The original order is often more neutral for “I’ll do X until Y happens.”

Is pysyn puistossa interchangeable with olen puistossa?

Not exactly:

olen puistossa = “I am in the park” (a state, doesn’t strongly emphasize staying)
pysyn puistossa = “I stay/remain in the park” (emphasizes continuing to stay and not leaving)

So pysyn fits especially well with until... because it highlights duration.