The number of Robins in Batman comics varies depending on different storylines and continuities. Generally, you can count at least four significant Robins: Dick Grayson, who was the first; Jason Todd, who had a more tumultuous story; Tim Drake, known for his detective skills; and Damian Wayne, Batman's son.
It would be a very dark and twisted take on the Batman story. The whole dynamic between Batman and Robin would be completely changed. Robin would likely be filled with rage and a desire for revenge against Batman. It would also shatter the image of Batman as a hero in the traditional sense.
Tim Drake could also be considered to have a top - notch origin. His detective - like skills in figuring out Batman's identity showed his unique approach. He wasn't driven by the same personal tragedy as Dick or Jason at first. Instead, he was more interested in the idea of Batman and Robin and how it was important for Gotham. His origin added a new dimension to the Robin concept with his cerebral nature.
There may or may not be such a 'batman kills robins parents fanfic'. Fanfics cover a wide range of ideas, some of which are quite out - there. However, this idea seems rather extreme for the Batman universe. Batman is usually a protector, and killing Robin's parents would be a huge departure from his normal values. If it exists, it's likely a very niche and perhaps controversial piece of fan - created work.