Learn Hardcore Greek: Οι γονείς μου νιώθουν μεγάλη περηφάνια για τα ελληνικά μου.

Νιώθουν is 3rd person plural, present tense of νιώθω, and it means to feel (an emotion or a physical sensation): they feel.
Αισθάνονται (from αισθάνομαι) also means they feel, and in this context you could say Οι γονείς μου αισθάνονται μεγάλη περηφάνια... with almost the same meaning.
In everyday speech νιώθω is a bit more common and sounds slightly more informal/natural; αισθάνομαι can sound a bit more formal or careful, but both are correct here.

Why does the sentence use the noun περηφάνια (pride) instead of the adjective περήφανοι (proud), like in English My parents are proud of my Greek?

Greek can express this idea either with a noun or with an adjective.
Here we have the noun construction:

νιώθουν μεγάλη περηφάνια = they feel great pride.

You can also say it with an adjective, which is closer to English:

Οι γονείς μου είναι πολύ περήφανοι για τα ελληνικά μου. = My parents are very proud of my Greek.

Both are natural. The noun version focuses a bit more on the feeling itself (pride), the adjective version on the state (being proud), but in practice they are interchangeable in this context.

What exactly does μεγάλη do in μεγάλη περηφάνια? Could I say πολύ περηφάνια instead?

Μεγάλη is an adjective meaning great / big, so μεγάλη περηφάνια is great pride.
You cannot say πολύ περηφάνια; that is ungrammatical because πολύ must agree in gender/number/case with the noun when it works as an adjective. The correct form would be:

πολλή περηφάνια = a lot of pride / much pride.

So you have two natural options:

μεγάλη περηφάνια → great (strong) pride, emphasizes intensity.
πολλή περηφάνια → a lot of pride, emphasizes quantity/extent.

Both are fine, just slightly different nuances.

Why is γονείς plural? Can I say ο γονιός μου instead?

Γονείς means parents (plural). In Greek, just like in English, it’s very common to talk about parents as a pair, so Οι γονείς μου = my parents.
You can use a singular form if you mean just one parent:

ο γονιός μου (informal: my parent)
ο πατέρας μου (my father)
η μητέρα μου (my mother)

So Οι γονείς μου naturally suggests you’re talking about both parents together.

Why does μου come after γονείς and ελληνικά, instead of before them like my parents or my Greek?

In Greek, the unstressed possessive pronouns (μου, σου, του, της, μας, σας, τους) normally come after the noun they modify:

οι γονείς μου = my parents
τα ελληνικά μου = my Greek (my Greek language skills)

This is simply the normal Greek pattern: article + noun + possessive pronoun.
If you want to emphasize possession, you can use stressed forms like δικός μου, but the everyday neutral way is to put μου after the noun.

Why do we still need the articles οι and τα if we already have μου (my)?

In Greek, when you use these unstressed possessive pronouns (μου etc.), you almost always keep the definite article:

οι γονείς μου (literally: the parents my)
τα ελληνικά μου (literally: the Greek (language) my)

The possessive μου by itself doesn’t replace the article the way my does in English. The combination article + noun + μου is what feels natural and complete in Greek.

Why is τα ελληνικά plural? Aren’t languages usually singular?

Names of many languages in Greek are neuter plural:

τα ελληνικά = Greek
τα αγγλικά = English
τα γαλλικά = French

Historically this came from meanings like Greek things / Greek words, and the plural form has stuck.
If you want a singular form, you typically say η ελληνική γλώσσα = the Greek language, but in everyday speech τα ελληνικά is what people use.

Why is ελληνικά not capitalized, while Greek in English is?

In Greek, names of languages are not capitalized (unless they are at the start of a sentence):

μιλάω ελληνικά, αγγλικά και γαλλικά.

So ελληνικά is written with a lowercase ε, even though in English Greek takes a capital G. This is just a difference in writing conventions between the two languages.

What does για do in για τα ελληνικά μου? Could I use another preposition or leave it out?

Για here means for / about, and it introduces what the pride is about:

νιώθουν περηφάνια για τα ελληνικά μου = they feel pride for my Greek.

With emotions like περηφάνια, Greek normally uses για to link the feeling to its object:

είμαι περήφανος για σένα = I’m proud of you.

You cannot omit για here, and other common prepositions (με, σε) would be wrong in this structure.

What grammatical case is περηφάνια in, and why?

Περηφάνια is in the accusative singular.
It is the direct object of the verb νιώθουν:

(Οι γονείς μου) – subject (nominative)
νιώθουν – verb
(μεγάλη) περηφάνια – direct object (accusative), the thing they feel.

So its case is determined by its role as the object of the verb.

Can I change the word order, for example:

Οι γονείς μου νιώθουν για τα ελληνικά μου μεγάλη περηφάνια, or
Μεγάλη περηφάνια νιώθουν οι γονείς μου για τα ελληνικά μου?

Yes. Greek word order is relatively flexible, and all of these versions are grammatically correct.
The different orders mostly change emphasis:

Οι γονείς μου νιώθουν μεγάλη περηφάνια για τα ελληνικά μου – neutral, balanced.
Οι γονείς μου νιώθουν για τα ελληνικά μου μεγάλη περηφάνια – extra focus on for my Greek before naming great pride.
Μεγάλη περηφάνια νιώθουν οι γονείς μου για τα ελληνικά μου – strong emphasis on great pride.

In normal speech, the original order is the most typical.

Is there any difference between περηφάνια and υπερηφάνεια?

Both mean pride and can often be used interchangeably.
Περηφάνια is the more everyday, colloquial form.
Υπερηφάνεια is a bit more formal / literary, and you’ll see it more in writing, speeches, or set expressions.
In this sentence, περηφάνια is exactly what most native speakers would say in conversation.

Could I replace νιώθουν μεγάλη περηφάνια with είναι πολύ περήφανοι? Would it mean the same thing?

Yes, that is a very natural alternative with almost the same meaning:

Οι γονείς μου είναι πολύ περήφανοι για τα ελληνικά μου.

Here:

είναι = they are
πολύ περήφανοι = very proud (adjective phrase)

The original focuses slightly more on the feeling (they feel great pride), while this version focuses on their state (they are very proud), but in everyday use both simply say that your parents are proud of your Greek.