Learn Hardcore Hungarian: Vasárnap a család a parkban tölti a napot, ez a kedvenc programjuk. - On Sunday the family spends the day in the park; this is their favourite program.

So vasárnap on its own already means on Sunday. You don’t say *on vasárnap or add anything in Hungarian; the time expression stands by itself at the start of the sentence.

Why is it a család tölti and not a család töltik? In English you might say the family spend (plural).

In Hungarian, család (family) is grammatically singular, and the verb always agrees with the grammatical number, not with the idea of “many people inside the family”.

a család tölti = the family spends (singular verb)
a családok töltik = the families spend (plural verb)

So tölti is 3rd person singular, matching a család (one family).

Why is the verb tölti and not just tölt? What does that -i ending mean?

Hungarian has two present-tense conjugations:

indefinite (used with no object or an indefinite object)
definite (used with a definite object: with a / az, a possessive, a proper name, etc.)

The verb is tölteni (to spend [time]).
For 3rd person singular:

Indefinite: ő tölt – he/she spends (something, not specifically defined)
Definite: ő tölti a napot – he/she spends the day

Because the object is a napot (the day – clearly definite), you must use the definite form:

✅ a család a parkban tölti a napot
❌ a család a parkban tölt a napot

So the -i in tölti signals the definite verb form.

Why is it a napot and not just a nap?

Hungarian marks direct objects with the accusative ending -t.

nap = day (subject or basic form)
napot = day as a direct object (the day)

In the sentence, napot is what is being spent:

tölti a napot = (he/she) spends the day

So you need the -t to show that napot is the object of the verb tölti.

What does a parkban literally mean, and what is the function of -ban?

A parkban breaks down like this:

a = the
park = park
-ban = in (inessive case ending)

So parkban literally means in (the) park. Inessive -ban / -ben is added to the noun instead of using a separate preposition:

házban = in the house
városban = in the city
parkban = in the park

The form -ban (not -ben) is chosen because park has a back vowel (a), and Hungarian vowel harmony applies.

Why does the sentence repeat the article a so often: a család a parkban tölti a napot?

In Hungarian, the definite article a / az is used in many places where English sometimes leaves out the.

a család = the family
a parkban = in the park
a napot = the day

English might say:
On Sunday, the family spends the day in the park.
Here we also actually use the three times, so in this case it matches well.

But in general, if something is specific or known from context, Hungarian usually marks it with a / az, even where English might allow dropping the.

Could the word order be different, like A család vasárnap a parkban tölti a napot? Does word order matter here?

Hungarian word order is flexible, but it is used to highlight focus and known/new information.

Your variants:

Vasárnap a család a parkban tölti a napot.
– Neutral, with vasárnap (on Sunday) as the first setting of the scene.
A család vasárnap a parkban tölti a napot.
– Also natural; here a család (the family) is introduced first, then vasárnap as the time.

Both are correct. In everyday speech, both would be understood the same way.
If a particular element is put right before the verb, that usually gets strong focus, but in this sentence nothing is in a strongly contrastive focus position, so multiple orders work.

What exactly does programjuk mean? It looks like program plus something.

Yes, programjuk is:

program = program, plan, activity, event
-juk = their (3rd person plural possessive suffix)

So programjuk = their program / their activity.

In this sentence:

ez a kedvenc programjuk = this is their favorite activity.

Because the possessors are “they” (the members of the family), you use the 3rd person plural suffix -juk.

A few related forms:

programom = my program
programod = your (singular) program
programja = his/her/its program
programjuk = their program (one program)
programjaik = their programs (more than one program)

Why isn’t there a form of to be (like van) in ez a kedvenc programjuk?

In Hungarian, in the present tense, the verb lenni (to be) is usually omitted in 3rd person (he/she/it/they) in simple “X is Y” type sentences.

So:

Ez a kedvenc programjuk.
Literally: “This their favorite program.”
Understood as: “This is their favorite program.”

If you write Ez van a kedvenc programjuk, it sounds wrong or at least very unnatural in this context.

You do use van / vannak:

when you need emphasis or contrast in certain structures
in non-present tenses
in existential sentences (e.g. Van egy könyvem. – I have a book / There is a book of mine.)

But a simple identification sentence in the present, 3rd person, usually drops van:

Ő az apám. – He is my father.
Ez a házunk. – This is our house.
Ez a kedvenc programjuk. – This is their favorite activity.

What part of speech is kedvenc here? Does it work like an adjective or a noun?

Kedvenc can be both an adjective and a noun in Hungarian.

As an adjective: kedvenc program = favorite program
As a noun: A kutya a kedvencem. = The dog is my favorite.

In ez a kedvenc programjuk, kedvenc functions as an adjective modifying programjuk:

kedvenc programjuk = their favorite program / activity.

So it’s “favorite” (adjective) + “their program” (possessive noun).

Does Vasárnap a család a parkban tölti a napot mean every Sunday or this Sunday?

By itself, vasárnap is a bit ambiguous. It can mean:

on Sunday (a specific Sunday, understood from context), or
on Sundays / on Sunday in general (habitual action), especially if the whole sentence describes a typical routine.

In many contexts, this sentence will be understood as habitual:

Vasárnap a család a parkban tölti a napot.
→ The family (typically) spends the day in the park on Sundays.

If you want to make the habitual meaning very clear, you can say:

Vasárnaponként a család a parkban tölti a napot. = On Sundays (each Sunday), the family spends the day in the park.

If you want to emphasize a specific coming Sunday, you would rely on context or additional words like most vasárnap (this Sunday) or a future-tense context.

Why is it ez a kedvenc programjuk and not az a kedvenc programjuk? When do you use ez vs. az?

Ez and az correspond roughly to this and that:

ez = this (near in space, time, or just mentioned)
az = that (farther away, or something more distant/other)

Here, the “this” refers to the activity just described in the first half of the sentence (spending Sunday in the park). Conceptually, that’s “this thing we’ve just been talking about”, so ez is natural:

…tölti a napot, ez a kedvenc programjuk.
→ …spends the day, this is their favorite program.

You could use az if you were contrasting with another, more distant option, but in standard neutral wording referring back to what you just said, ez is the default.