Çocuklar kaykay parkurunda özgürce kayıyor.

parkurunda breaks down into three pieces:

1. parkur (park/course)
2. -u (3rd person singular possessive, “its/the”)
3. -nda (locative case, “in/at”)

So parkurunda literally means “in the skate­park” (a specific skatepark). If you drop the possessive -u, you get parkurda, which is simply “in a skatepark” (indefinite).

Turkish does not use articles like a/the in the same way English does. Indefinite nouns generally appear without bir unless you want to emphasize “one.” To say “in a skatepark,” you simply use the locative on parkur without the possessive:
• kaykay parkurda = “in a skatepark”

If you really wanted to stress “one skatepark,” you could add bir:
• bir kaykay parkurda

The original sentence omits bir because it’s clear from context that it’s just “the local skatepark” or “a skatepark” in general.

özgür is an adjective meaning “free.” To turn it into a manner adverb (“freely”), Turkish adds -ca/-ce. Vowel harmony dictates the front form -ce after a front vowel like ü:
• özgür + -ce → özgürce (“freely”)

Adjectives in Turkish (like özgür) only modify nouns. To modify a verb (here kaymak → to skate/slide), you must use an adverb. özgürce tells us how the action happens (“freely”).

The verb root is kay- (“to slide/glide/skate”). To create the present-continuous tense, you add -(i)yor:
• kay + ıyor → kayıyor (“is skating/sliding”)

Since a is a back vowel, we use -ıyor (not -iyor). Turkish does not attach an extra ending for 3rd person singular here, so kayıyor means “he/she/it is skating.”

True, kaymak literally means “to slide” or “glide.” In everyday Turkish kaymak covers sliding, skating, skiing, surfing, etc. When you talk about riding a skateboard, you simply use kaymak with kaykay (skateboard) to get “to skateboard.”

Turkish word order is fairly flexible, but the neutral pattern for a simple sentence is:
Subject – Place – Manner – Verb

In our example:
Çocuklar (subject)
kaykay parkurunda (place)
özgürce (manner)
kayıyor (verb)

You can shift elements for emphasis, but placing the adverb right before the verb is the most common and clear way to say “they are skating freely.”