Because both forms of plankton spend the daylight hours in the top meter of the ocean, oil, which tends to float, will affect them directly and dramatically.
Phytoplankton, which are photosynthetic, plant-like forms of plankton, need to receive sunlight in order to survive. An oil spill will coat both the plankton and the water's surface, blocking off sunlight and starving the plankton.
Zooplankton, which must eat to survive, will also have issues with the oil spill. Most zooplankton move about by using flagella or cilia, which are very fine hairlike structures. If these structures get coated by oil, the plankton will not be able to move and will starve.