There are two things to consider about the role of genetics in parallel evolution.
The first is that the genetic code for a given species may contain the potential for many complex structures that aren't actually expressed in that organism. Imagine a construction crew building a house. The blueprint may contain the instructions to build an addition on the back of the house, but unless the architect tells the crew to build that part, they'll only build the basic house, without the addition. Our genetic equivalent to the architect would be another mutation that activates the portion of the DNA needed to actually express a trait.
Jellyfish and anemones are animals with a radial body plan -- they have no left or right side. However, their genetic code has been found to contain a marker for a bilateral body plan [source: Ars Technica]. For some reason, it isn't expressed in members of the jellyfish family.
Why is this important for parallel evolution? It shows that very primitive organisms can have the genetic tools available to create greater complexity. As the organism evolves, widely separated species can develop similar traits because the potential for those traits was there right from the beginning.
The second thing to consider is the experimental evidence. Recently, biologists have gone beyond morphology in their examination of parallel evolution. They have found proof that in at least some cases, morphological similarities were matched by genetic similarities. The chemical interactions of proteins and amino acids that cause the morphological changes were also the same in two species that had been isolated from each other for millions of years [source: ScienceDaily].
If you want to learn more about evolution, natural selection and animals, try the next page.