Σήμερα περνάω καλά με την παρέα μου στο πάρκο.

The verb περνάω literally means I pass (as in I pass time, I spend time).
The adverb καλά means well.

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

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

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.

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.

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.

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.

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.

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.

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.

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.