"The red flower blooms while the water is flowing."
Translation:물이 흐르면서 빨간 꽃은 핍니다.
호르다 is the verb of flowing, and the form 호르면서 is creating the if->then aspect of the sentence. The 면 means 'if', and 서 cues that the next part is the result of the first.
"흐르면서" from the verb "흘리다"="to flow" It can be also used with the word blood "피" to be "피를 흘리다"="to bleed"
I thought so too. As another commenter pointed out, when one sentence modifies another the main sentence tends to have the topic marker while the modifying sentence gets the subject marker.