Learn Hardcore Greek: Σήμερα περνάω καλά με την παρέα μου στο πάρκο. - Today I am having a good time with my group of friends in the park.

So literally περνάω καλά is I pass (my time) well, i.e. I’m spending my time well.
Idiomatic English expresses this as I’m having a good time or I’m enjoying myself.

You can change the adverb for different nuances:

περνάω άσχημα = I’m having a bad time
περνάω υπέροχα = I’m having a wonderful time

Why is it καλά and not καλός after περνάω?

Καλός is an adjective (good) and agrees with nouns:

καλός φίλος = good (male) friend
καλή μέρα = good day
καλό παιδί = good child

Καλά here is an adverb (well) and it modifies the verb περνάω.
So:

περνάω καλά = I pass (time) well → I’m having a good time

You use καλά with verbs:

νιώθω καλά = I feel well
κοιμάμαι καλά = I sleep well

What is the difference between περνάω and περνώ?

They are two forms of the same verb. Both mean I pass / I spend (time).

περνάω is more common in everyday spoken Greek.
περνώ sounds a bit more formal or literary but is also correct.

In the present tense, both sets are possible:

εγώ περνάω / περνώ
εσύ περνάς
αυτός/αυτή/αυτό περνάει / περνά
εμείς περνάμε
εσείς περνάτε
αυτοί περνάνε / περνούν

In your sentence, περνάω is simply the colloquial choice.

Why is it την παρέα μου and not something like η μου παρέα?

Greek normally puts the possessive pronoun after the noun, as a clitic:

η παρέα μου = my group of friends / my company
το βιβλίο σου = your book
το σπίτι του = his house

So the usual order is: article + noun + possessive pronoun

In your sentence, με την παρέα μου means with my group of friends / with my company.
Η μου παρέα is not idiomatic modern Greek.

What exactly does παρέα mean? Is it the same as friends?

Παρέα is a feminine noun and literally means company, group of people you hang out with, or the act of being together socially.

In με την παρέα μου, it usually implies:

my group of friends
the people I’m hanging out with

Nuance:

με τους φίλους μου = with my friends (emphasis on the people as friends)
με την παρέα μου = with my group / my crowd (emphasis on the social group / togetherness)

Often, η παρέα μου in context is effectively my friends, but with a slightly more social / informal flavor.

Why is the article feminine (την) in την παρέα μου?

Because παρέα is a feminine noun. Its basic forms:

η παρέα (nom. sg.) = the group / the company
της παρέας (gen. sg.)
την παρέα (acc. sg.)

After the preposition με (with), Greek uses the accusative case, so:

με την παρέα μου = with my group / with my company

Hence την (feminine accusative) is required.

Why do we use μου after παρέα and not a separate word like δικός μου?

Μου here is the weak (clitic) form of εγώ used for possession: my.

The normal, neutral way to say my X is:

ο φίλος μου = my friend
η παρέα μου = my group (of friends)
το πάρκο μου = my park

Forms like δικός μου, δική μου, δικό μου add emphasis, meaning my own:

η δική μου παρέα = my own group (as opposed to someone else’s)

In this sentence, simple possession is enough, so η παρέα μου.

What does στο πάρκο mean exactly, and how is it formed?

Στο is a contraction of:

σε (in, at, to) + το (the, neuter singular)

So:

σε + το πάρκο → στο πάρκο = in the park / at the park / to the park (depending on context)

Greek very often contracts σε + definite article:

σε + το → στο
σε + τον → στον
σε + την → στη(ν)
σε + τους → στους

In your sentence, στο πάρκο most naturally means at the park or in the park.

Can the word order change? For example, can I say Περνάω καλά σήμερα με την παρέα μου στο πάρκο?

Yes. Greek word order is quite flexible. All of these are possible and natural, with slightly different emphasis:

Σήμερα περνάω καλά με την παρέα μου στο πάρκο.
→ neutral; emphasizes today first.
Περνάω καλά σήμερα με την παρέα μου στο πάρκο.
→ slightly more focus on I’m having a good time, then specifying today.
Με την παρέα μου περνάω καλά σήμερα στο πάρκο.
→ puts more emphasis on with my group as the reason I’m having a good time.

The basic structure (subject–verb–adverb–complements) is clear in all of them, and context + intonation carry the nuance.

Does the Greek present tense περνάω correspond to English I pass or I am passing?

Greek present tense usually covers both simple present and present continuous meanings in English.

So περνάω καλά can mean:

I have a good time (in general / habitually)
I am having a good time (right now)

In this sentence, because of σήμερα (today), the natural interpretation is:

Today I’m having a good time with my friends at the park.

Context tells you whether it’s general or about right now.