Annak ellenére, hogy nincs mentesség, a várakozási díj alacsony.

It is a fixed expression meaning despite the fact that, in spite of the fact that, or sometimes even though.

In this sentence:

• annak ellenére = despite that
• hogy = that, introducing a clause

So Annak ellenére, hogy nincs mentesség... means Despite the fact that there is no exemption...

Annak is the form of az (that) used with many postpositions.

So:

• annak ellenére = despite that
• ennek ellenére = despite this

When Hungarian wants to say despite the fact that..., it keeps this annak and then adds a hogy clause:

• annak ellenére, hogy...

This is very natural Hungarian.

Because annak ellenére can be followed either by a noun phrase or by a whole clause.

Compare:

• A tiltás ellenére elment. = He went despite the ban.
• Annak ellenére, hogy tilos volt, elment. = He went despite the fact that it was forbidden.

Here, nincs mentesség is a full clause, so Hungarian introduces it with hogy.

Because nincs is the normal negative form of van in existential or there is/there are sentences.

So:

• Van mentesség. = There is an exemption.
• Nincs mentesség. = There is no exemption.

Standard Hungarian does not use nem van here.

In van/nincs sentences, an indefinite noun often appears with no article.

For example:

• Van idő. = There is time.
• Nincs pénz. = There is no money.
• Nincs mentesség. = There is no exemption.

So mentesség here is being presented as something absent in general, not as a specific already-mentioned item.

Hungarian usually leaves out the present-tense copula (is / are) in the 3rd person when the predicate is a noun or adjective.

So:

• A várakozási díj alacsony. = The parking/waiting fee is low.

But in other persons or tenses, the verb appears:

• Alacsony vagyok. = I am low/short.
• A díj alacsony volt. = The fee was low.

This is one of the most important differences from English.

Literally, it is something like waiting fee, but in many real-world contexts it means parking fee.

The parts are:

• várakozás = waiting; in traffic/official language, often parking
• várakozási = relating to waiting/parking
• díj = fee

So the most natural English translation often depends on context. On signs or in regulations, parking fee is often the best choice.

The ending -i turns a noun into an adjective-like modifier.

So:

• várakozás = waiting / parking
• várakozási = waiting-/parking-related

That lets it modify another noun:

• várakozási díj = parking fee / waiting fee

This -i is very common in Hungarian.

Not especially. It is a very normal, neutral structure:

• Annak ellenére, hogy nincs mentesség, = introductory subordinate clause
• a várakozási díj alacsony. = main clause

Hungarian word order is flexible, but this version is straightforward and natural, especially in formal or written language.

Because Hungarian separates subordinate clauses from the main clause with commas.

Here the concessive clause comes first:

• Annak ellenére, hogy nincs mentesség,
• a várakozási díj alacsony.

So the commas help mark the clause boundary. This is standard punctuation.

Yes. That would also be correct.

• Bár = although / even though
• noha = although
• annak ellenére, hogy = despite the fact that

So these are all possible, but they feel slightly different:

• Bár nincs mentesség... = more compact
• Annak ellenére, hogy nincs mentesség... = a bit more explicit and formal

Both are natural Hungarian.

It is a noun meaning exemption, freedom from something, or in some contexts immunity.

It is related to:

• mentes = free from, exempt
• mentesség = exemption

So:

• adómentes = tax-free
• adómentesség = tax exemption

In your sentence, mentesség most likely means some kind of fee exemption or waiver.