This is a really good question and something a lot of people usually don't think about. The answer is that all plant cells need oxygen to live, because without oxygen they can't perform aerobic respiration (respiration is the process of breaking down food to get energy). Of course you probably know that when plants perform photosynthesis, they combine water, carbon dioxide, and the sun's energy to produce sugar and oxygen. So the cells in the green parts of the plant, where photosynthesis is taking place, get all the oxygen they need from the oxygen produced by photosynthesis. So cells in the leaves and stems are okay. The trick is the cells down in the roots, where there is no photosynthesis.

In most plants, these cells get their oxygen from air in the spaces between dirt particles in the soil (you'd be surprised how much empty space there is in the soil -- mostly because earthworms are always moving around, churning up the dirt).