Parkta yürüyebilmek için spor ayakkabı giymeliyiz.

The suffix -ebil is the ability/modality suffix. It attaches to the verb root to mean “be able to.”

• yürü- = “to walk”
• -ebil = “be able to”
  Together you get yürü-yebil (“be able to walk”).

-mek (or -mak) is the infinitive suffix in Turkish.
When you want to turn a verb into a noun (so you can use it with için, postpositions, or other constructions), you need the infinitive. Hence yürü-yebil-mek = “to be able to walk.”

In Turkish, to express purpose (“in order to…”) you put için after a nominalized verb (infinitive).

• yürüyebilmek = “to be able to walk”
• yürüyebilmek için = “in order to be able to walk”

parkta is the locative case, marked by -ta/-te, meaning “in/at the park.”

• parkta yürümek = “to walk in the park.”
  parka is the dative case, marked by -ya/-ye, meaning “to the park.”
• parka yürümek = “to walk to the park.”

giymeliyiz breaks down as:

• giye- = root of giymek (“to wear”)
• -meli = necessity/obligation suffix (“must/should”)
• -yiz = 1st person plural ending (“we”)
  So giymeliyiz = “we must/should wear.”

In Turkish, when talking about something in general (like a type of shoe), you often use the singular indefinite form. It’s like saying “We must wear sports shoes” but without marking plural. You could say spor ayakkabılar giymeliyiz, and it’s grammatically correct, but the singular form is more common for general statements.

Turkish is a pro-drop language: subject pronouns are often omitted because the verb ending (-yiz here) already tells you the person (we). Including biz is possible for emphasis, but not necessary.