Ystäväni on kiitollinen vanhemmilleen, koska he tukevat hänen opintojaan.

Ystäväni consists of:

• ystävä = friend
• -ni = my (1st person singular possessive suffix)

So ystäväni literally means my friend.

In Finnish, possession is often shown with a suffix instead of (or in addition to) a separate pronoun. You can also say:

• minun ystäväni = my friend

Here minun is the separate pronoun my, and -ni repeats the possession. Both ystäväni and minun ystäväni are correct; using just ystäväni is slightly more compact and very common.

The verb olla (to be) conjugates like this in the present tense:

• minä olen – I am
• sinä olet – you (sg) are
• hän on – he/she is
• me olemme – we are
• te olette – you (pl) are
• he ovat – they are

The subject of the sentence is Ystäväni = my friend. That is a 3rd person singular subject (like hän), so the verb must also be 3rd person singular:

• Ystäväni on kiitollinen… = My friend is grateful…

Using olen would mean I am grateful…, which would not match the subject Ystäväni.

Kiitollinen is an adjective meaning grateful.

In this sentence it is a predicative adjective linked to the subject by olla:

• Ystäväni on kiitollinen = My friend is grateful.

For a singular subject, the predicative is in the nominative singular:

• Ystäväni on kiitollinen.

If the subject were plural, the adjective would also be plural partitive:

• He ovat kiitollisia. = They are grateful.

So here kiitollinen is singular because Ystäväni is one person.

Vanhemmilleen roughly means to his/her parents (or to their parents depending on context). It is built like this:

1. vanhempi = parent (also “older” as an adjective)
2. plural nominative: vanhemmat = parents
3. allative plural stem: vanhemmille = to the parents
   
   • -lle is the allative ending: direction towards a person/thing, often “to” someone.
4. possessive suffix -en = his/her/their

So:

• vanhemmille = to the parents
• vanhemmilleen = to his/her parents (or their parents)

The verb/adjective combination here is:

• olla kiitollinen jollekin = to be grateful to someone

That’s why vanhemmilleen is in the allative (-lle) case.

Koska is a conjunction meaning because, introducing a reason clause:

• …, koska he tukevat hänen opintojaan.
  = …, because they support her studies.

In standard written Finnish, a comma is normally used before a subordinate clause introduced by words like koska, että, vaikka etc.

You can also move the koska-clause to the beginning:

• Koska he tukevat hänen opintojaan, ystäväni on kiitollinen vanhemmilleen.

The comma stays, but now it comes after the subordinate clause.

He is the 3rd person plural pronoun = they.

It refers back to vanhemmilleen (his/her parents). So:

• vanhemmilleen → his/her parents
• he tukevat → they support

So the second clause means: because *they (the parents) support her studies*.

The dictionary form tukea is the infinitive “to support”.

The present tense forms of tukea are:

• minä tuen – I support
• sinä tuet – you (sg) support
• hän tukee – he/she supports
• me tuemme – we support
• te tuette – you (pl) support
• he tukevat – they support

Because the subject is he (they), we use tukevat, the 3rd person plural present tense form:

• he tukevat = they support

Hänen is the 3rd person singular possessive pronoun = his/her.

Although Ystäväni contains “my”, it refers to the friend as a 3rd person (he/she), not as me. The sentence means:

• My friend is grateful to her parents because they support *her studies.*

So:

• Ystäväni = my friend
• hänen opintojaan = her studies (belonging to that friend)

If you said minun opintojani, it would mean my studies (the speaker’s), which would change the meaning:

• Ystäväni on kiitollinen vanhemmilleen, koska he tukevat minun opintojani.
  = My friend is grateful to her parents because they support *my studies.*

Opintojaan can be broken down like this:

1. opinto = a study (as in university studies)
2. plural partitive: opintoja = studies (some studies, her studies in general)
   
   • -ja here marks plural partitive.
3. possessive suffix: opintoja
   • -n → opintojaan

The -n at the end is the 3rd person possessive suffix, showing that the studies belong to him/her:

• opintoja = studies
• opintojaan = his/her studies

So hänen opintojaan means her studies (in the partitive plural).

The verb tukea (to support) often takes the partitive object when the action is:

• ongoing / not completed, or
• affecting something in an open, not-bounded way, or
• talking about something abstract or uncountable in general.

Compare:

• he tukevat hänen opintojaan = they support her studies (in general, ongoing)
• he tukevat hänen opintonsa – would sound like supporting a specific, more delimited set of studies; not the usual way to say this.

So opintojaan is:

• partitive plural = opintoja
• plus possessive suffix -n = his/her

This fits the natural Finnish expression for supporting (someone’s) studies as an ongoing process.

Yes, Finnish allows flexible word order for emphasis.

Some natural variants:

1. Original:

   • Ystäväni on kiitollinen vanhemmilleen, koska he tukevat hänen opintojaan.

2. Koska-clause first (emphasizing the reason):

   • Koska he tukevat hänen opintojaan, ystäväni on kiitollinen vanhemmilleen.

3. Moving vanhemmilleen:

   • Ystäväni on vanhemmilleen kiitollinen, koska he tukevat hänen opintojaan.
     (more emphasis on to her parents)

All of these are grammatically correct; the differences are mainly in focus and emphasis, not in basic meaning.

Both are possible, but they express slightly different ideas:

1. kiitollinen jostakin (+ -sta/-stä, elative)

   • Ystäväni on kiitollinen vanhemmistaan.
     = My friend is grateful *for her parents.
     Here the parents themselves are the *thing she is grateful for.

2. kiitollinen jollekin (+ -lle, allative)

   • Ystäväni on kiitollinen vanhemmilleen.
     = My friend is grateful *to her parents.
     Here the parents are the *people to whom she feels gratitude (typically for what they have done).

In your original sentence, vanhemmilleen fits very well, because the next clause explains what they do:
they support her studies, and she is grateful to them for that.