Parka gitmek için bu köprüden geçmeliyiz.

Because -a / -e is the dative case, which often means to or toward.

• park = park
• parka = to the park

In this sentence, parka gitmek means to go to the park.

The form is -a here because park has a back vowel, so vowel harmony gives -a rather than -e.

Gitmek is the infinitive form of gitmek = to go.

So:

• gitmek = to go
• gitmek için = in order to go / for going

This is a very common Turkish structure:

• verb + -mek / -mak + için

It expresses purpose.

Examples:

• öğrenmek için = in order to learn
• görmek için = in order to see
• parka gitmek için = in order to go to the park

Because için is a postposition in Turkish, not a preposition.

In English, we put words like for and to before another word or phrase. In Turkish, words like için usually come after the word or phrase they belong to.

So Turkish says:

• gitmek için = literally to go for
• natural English meaning: in order to go

That is normal Turkish structure.

Bu köprüden means from this bridge, but with geçmek it often has the practical meaning through / via / over this bridge.

Breakdown:

• bu = this
• köprü = bridge
• köprüden = from the bridge / via the bridge

The suffix -den / -dan / -ten / -tan is the ablative case. It often means from, but Turkish uses it in some expressions where English would say through, via, or by way of.

So in this sentence, bu köprüden geçmeliyiz means something like:

• we must go across by using this bridge
• we must pass over this bridge

Both are possible in Turkish, but they can suggest slightly different things.

• köprüyü geçmek = to cross the bridge
• köprüden geçmek = to pass through/by way of the bridge, to go over it

In everyday use, both can refer to crossing a bridge, but köprüden geçmek sounds very natural when the bridge is the route you use to get somewhere.

So in this sentence, bu köprüden geçmeliyiz focuses on using this bridge as the way to go.

Geçmeliyiz comes from the verb geçmek = to pass, to go through, to cross.

It breaks down like this:

• geç- = pass / cross
• -meli / -malı = must / should
• -yiz = we are / we

So geçmeliyiz means we must pass or we should cross.

The -meli / -malı suffix expresses necessity, but depending on context it can sound like:

• must
• have to
• should

Here, must is a very natural translation.

The y is a buffer consonant.

The necessity suffix -meli / -malı ends in a vowel, and the personal ending also begins with a vowel sound. Turkish often inserts y between vowels to make pronunciation smoother.

So:

• geçmeli + iz would be awkward
• Turkish uses geçmeliyiz

This kind of buffer consonant is very common in Turkish.

It is built into the verb geçmeliyiz.

Turkish often leaves subject pronouns out because the verb ending already shows the person.

Here, -yiz tells us the subject is we.

So:

• geçmeliyim = I must pass
• geçmelisin = you must pass
• geçmeli = he/she/it must pass
• geçmeliyiz = we must pass

Because of that, Turkish does not need a separate biz unless the speaker wants extra emphasis.

Because Turkish normally prefers Subject-Object-Verb order, and the main verb often comes at the end.

So this sentence is arranged in a very typical Turkish way:

• Parka gitmek için = in order to go to the park
• bu köprüden = via this bridge
• geçmeliyiz = we must cross

A very literal order would be:

• To the park go in order to, this bridge via, we must pass

That sounds strange in English, but it is normal in Turkish.

Yes. -meli / -malı can express different levels of necessity depending on context.

It can mean:

• must
• have to
• should

So geçmeliyiz could be understood as:

• we must cross
• we should cross

In this sentence, if the bridge is the necessary route to the park, must is probably the best choice. If it is more like advice, then should could also fit.