Who cook the soup? Confusion
The fundamental sentence is 他煮湯 (he cooks soup). The subject is 他 and the object is 湯, or the thing that is cooked.

You may consider 第一次 (the first time) and 為我 (for me) as adverbial phrases. They describe how the soup was cooked by Father.

為+sb, 替+sb, and 幫+sb indicate the action is done on behalf of someone. They behave like 給+sb. E.g.:

我為你打掃了房間 I cleaned the room for you.
他替我做功課 He does homework for me.
我幫她買了雞蛋 I bought eggs for her.