Let's not confuse parthenogenesis with hermaphroditic reproduction! They're not the same at all. In parthenogenesis, the female gamete does all the work single-handedly. Hermaphrodites, however, are remarkable for being able to produce both male and female gametes. That's because they have both male and female reproductive organs.
In some species, like certain flowers, snails and fish, hermaphroditism is common. Sometimes an organism might start off male and turn female; sometimes they might start off female and turn male; and sometimes they remain both sexes throughout their lifespan.
Remember Nemo and his dad in "Finding Nemo"? It turns out a sequel is warranted because there's a lot of drama in the reproductive lives of clownfish. Socially, clownfish like to hang out in sea anemones where two large breeders, a male and a female, are in charge of a group of smaller male clownfish who don't reproduce.
But say, for instance, the female breeder dies. In that case, the large male switches its sex to become the dominant female while the biggest of the small males quickly bulks up to take on the role of sexually mature male breeder [source: UCB]. Come on Pixar — "Becoming Nema" is a hit waiting to happen.