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”)