I saw this title not long ago in a Bloomberg "breaking news" banner. I immediately opened and read it voraciously. How, is this possible? a war scenario in the middle east resulting in lower oil prices? After dumping this "clickable" headline, the article rapidly moves to say the risk of a closure of the Gulf of Aden might be on the low side, mostly because of an American naval base on the other side of the Strait in Djibouti. Basically, the article is saying, there is a risk of escalation, but maybe not, and therefore oil prices will fall? Nonsense. The headline is misleading, a fighting in the middle east might lead to "nothing" or and "increase" on oil prices, but definitively not to "lower oil prices". A comprehensive Iran nuclear deal with lifting of sanctions can lead to a lower oil price. That is a different headline, but I guess it will not be clicked so many times.
Yemen fighting, can potentially lead to a huge rally in oil prices if the war becomes highly asymmetrical. In the event of a ground operation from the Saudi led army with air support, it might force the other side to quickly adopt guerrilla warfare tactics to palliate the losses. This will target valuable assets to the Saudis, most likely, pipelines, or whatever is thought of value to them. But, there is an scenario where a war can cause lower oil prices. In the event large refineries get attacked, there could be less upside to prices or even a small drop, as more crude will need to find other home (assuming production is not affected). Even then, this is a long shot since the biggest refineries Yambu and Jubail, are far from the border with Yemen. So, to conclude, no, I don't think Yemen fighting can lead to lower oil prices, certainly not because of the reasons stated in the Bloomberg article.