Learn Hardcore Hungarian: Ma nem a parkba megyek, hanem a könyvtárba. - Today I am not going to the park but to the library.

Ma nem a parkba megyek, hanem a könyvtárba.
→ The focus is on a parkba (the destination).
→ Meaning nuance: It’s not the park that I’m going to today, but the library.

Compare with:

Ma nem megyek a parkba, hanem otthon maradok.
→ Focus is on megyek (the action of going).
→ Meaning: Today I’m not going to the park at all, instead I stay at home.

So in your sentence, nem directly negates the place phrase a parkba, not the action of going itself. That’s why nem comes right before the focused phrase + verb:
nem a parkba megyek.

Could I also say Ma nem megyek a parkba, hanem a könyvtárba (megyek)? Does it mean the same thing?

Yes, you can, and the meaning is very close, but the emphasis is slightly different.

Ma nem a parkba megyek, hanem a könyvtárba.
→ Emphasis on which place you go to (park vs. library).
→ Nuance: Today my destination is not the park but the library.
Ma nem megyek a parkba, hanem a könyvtárba megyek.
→ Emphasis more on the whole action of going to the park vs. going to the library.
→ Nuance: Today I’m not going to the park; instead I’m going to the library.

In everyday speech, both are perfectly natural. The first version is slightly more focused on contrasting the two places; the second sounds more like contrasting two different complete actions. For communication, they’re effectively equivalent here.

What exactly does the ending -ba in parkba and könyvtárba mean?

The ending -ba / -be is a case ending that usually means into or to the inside of something.

park → parkba = (into) the park / to the park
könyvtár → könyvtárba = (into) the library / to the library

In English, we often just say to the park / to the library, but in Hungarian the direction is encoded in the noun by this suffix.

Choice between -ba and -be is based on vowel harmony:

Words with mostly back vowels (a, á, o, ó, u, ú) → -ba
- park → parkba
- iskola → iskolába
Words with only front vowels (e, é, i, í, ö, ő, ü, ű) → -be
- kert → kertbe
- büfé → büfébe

Könyvtár has both front (ö) and back (á) vowels; Hungarian looks at the last vowel, which is á (back), so it takes -ba: könyvtárba.

What’s the difference between -ba / -be and -ban / -ben? I see parkba here, but sometimes I see parkban.

-ba / -be = into / to the inside (direction, movement)
- A parkba megyek. – I’m going to (into) the park.
-ban / -ben = in / inside (location, no movement)
- A parkban vagyok. – I’m in the park.

So in your sentence, because the verb megyek expresses movement, you need the directional case -ba:
a parkba megyek, not a parkban megyek.

Why is the article a used before both parkba and könyvtárba? Could I leave it out?

In the sentence:

a parkba
a könyvtárba

the article a is the definite article (like the in English).

Using the article usually suggests a specific or known place:

Ma nem a parkba megyek, hanem a könyvtárba.
→ Implies the park and the library that both speaker and listener can identify (e.g. the local park and the usual library).

You can omit the article:

Ma nem parkba megyek, hanem könyvtárba.

This sounds a bit more generic or loose: not to a park but to a library (some park, some library).
With places, Hungarians quite often include the article if they have a specific one in mind, so your original version with a is very natural.

Why is the verb megyek only in the first part? Why not repeat it after hanem?

Hungarian often omits repeated elements when they are obvious from context.

The full, completely explicit version would be:

Ma nem a parkba megyek, hanem a könyvtárba megyek.

Because megyek is the same in both parts, it is natural to drop it in the second clause:

Ma nem a parkba megyek, hanem a könyvtárba.

Native speakers feel the missing megyek automatically:
“…not to the park, but to the library (I go).”

Repeating the verb is not wrong; it just sounds a bit heavier or more emphatic.

What is the role of hanem here, and how is it different from de?

Hanem is used for a correcting contrast after a negation. It corresponds to but rather / but instead in English.

Pattern: nem X, hanem Y = not X, but (rather) Y.

Ma nem a parkba megyek, hanem a könyvtárba.
→ Today I’m not going to the park, but rather to the library.

Key points:

hanem almost always comes after nem (or another negative word).
It contrasts the exact thing that was negated with another option.

De is a general but, however, not tied to a negation:

Ma a könyvtárba megyek, de fáradt vagyok.
→ Today I’m going to the library, but I’m tired.

You wouldn’t normally say:

✗ Ma nem a parkba megyek, de a könyvtárba.

In this structure, hanem is the correct choice.

Why is the subject pronoun én (I) missing? Where is the “I” in this sentence?

Hungarian is a pro‑drop language: the subject pronoun is often left out because the verb ending shows who the subject is.

megyek = I go / I am going
mész = you (singular) go
megy = he / she / it goes
megyünk = we go
mentek = you (plural) go
mennek = they go

So:

Ma nem a parkba megyek, hanem a könyvtárba.
→ The -ek ending on megyek already tells us it’s 1st person singular (I).

You could add én for extra emphasis:

Ma nem a parkba én megyek, hanem a könyvtárba.
or more naturally:
Én ma nem a parkba megyek, hanem a könyvtárba.

Then you are stressing I (as opposed to someone else).

Can I move ma to another position, like Nem ma a parkba megyek? How does that change the meaning?

Yes, you can move ma (today), but Hungarian word order changes what is emphasized.

Ma nem a parkba megyek, hanem a könyvtárba.
→ Normal, neutral topic: ma is just giving the time.
→ Emphasis: not to the park but to the library (today).
Nem ma a parkba megyek, hanem holnap.
→ It’s not today that I’m going to the park, but tomorrow.
→ Here you are negating the time of the action, not the place.

General idea:

Anything right before the verb (or before nem+verb) tends to be the focus.
Moving ma closer to the focus position (just before the verb) can shift emphasis to when instead of where.

Why is the present tense megyek used, even though in English we’d say “I’m not going to the park today, but to the library” (future meaning)?

Hungarian commonly uses the simple present tense for near future plans, especially with a time expression like ma (today), holnap (tomorrow), etc., and with verbs of movement like menni (to go).

Ma nem a parkba megyek.
Literally: Today I don’t go to the park.
Natural English: Today I’m not going to the park.

You could use a future form:

Ma nem a parkba fogok menni, hanem a könyvtárba.

But that sounds more formal or more emphatic, often used when you’re making a strong statement or prediction. In everyday speech, present tense + time adverb is the most natural way to talk about scheduled or planned future activities.

Is there any difference between Ma nem a parkba megyek, hanem a könyvtárba and Ma nem a parkba, hanem a könyvtárba megyek?

Both are correct and very natural. The difference is subtle and mostly about rhythm and focus.

Ma nem a parkba megyek, hanem a könyvtárba.
→ Focus is on a parkba as the negated destination.
→ The verb megyek is grouped more tightly with a parkba.
Ma nem a parkba, hanem a könyvtárba megyek.
→ The contrast nem a parkba, hanem a könyvtárba is presented as a pair, and megyek comes after both.
→ Feels like: I’m going not to the park but to the library.

In practice, speakers use both; the meaning is effectively the same: you’re contrasting the two destinations. The first is slightly more typical in textbooks, the second is very natural in spoken Hungarian.