Your mitochondrial DNA is the same as your mothers mitochondrial DNA. The reason for this is that mitchondria were present in the original egg cell when it formed and were present in every cell division since then after it was fertilized. I think typically there are like 100 in each cell and so when a cell divides about 50 go one way and 50 the other. As the cells grow the mitochondria do their own division to keep up. The only way the mitochondrial DNA can change is through mutation which isn't all that often.
What the theory is for this parasite is that the mermaid physiology is largely expressed by a similar mechanism. This allows the physiology to be inherited from only the mother.
I guess there is a issue when speaking about the mermaid 'mother' as a single entity when in reality they are two organisms in a symbiotic relationship. I'd say since the lower body is entirely replaced with the parasitic fish's tail, this would include the sexual organs.
Basically the vagina and ovaries of a mermaid originate from the parasite while the host's original genitalia and reproductive structures are digested. But even though these structures came from the fish half, they are still connected to the hosts nervous system which makes them fully functional.
Likely during spawning season the parasite floods the host's blood stream with endorphins which increases the host's libido and causes the host to seek out a reproductive mate.
Now once the parasite's eggs are fertilized by a human semen they must either be carried to term in a uterus (human style) or oviposited somewhere (fish style). I'm not sure which is more likely... Do the large breasts that grow on the human half actually lactate? If they do then that would suggest the mermaids care for their young after they are born but in this case the young are just parasitic fish that won't become mermaids in their own right until they find a host to infect.