Why did Sting and Rogue kill their father dragons? Is it to become third-generation dragon slayers, or what?
MASSIVE SPOILER ALERT
I apologize for not able to give any specific details or chapters of where it's from, but in the manga of Fairy Tails, when Natsu's team first faced off Sting and Rogue, they were informed that Sting and Rogue killed their dragons to become a proper Dragon Slayer. Later on when it's the crisis of magic disappearing (with all the giant faces), 5 Dragons showed up and saved the day. It was then that
Shadow dragon and the white dragon told everyone that they encouraged Sting and Rogue to kill themselves because 1. they were sick anyway, 2. to hide their existence from Acnologia (the Dragon Slayer who became a Dragon himself), 3. hopefully they would become stronger and more confident from killing them.
Chapter 413 shows the dragon destroying the faces and if you read from around there you should be able to find what you're looking for. The information I have given might be a bit off but hope it helps.
In many martial arts, Asian culture films, anime and stories, you see the student always trying to beat their master, proof they have surpassed the teacher. In this case they go one step further. It's a tradition of sorts and usually the troubled or evil will go that extra step. Always tradition in these stories or a variation of tradition.
Originally born four hundred years ago, Sting, an orphan, was raised by the Dragon Weisslogia, who taught him Light Dragon Slayer Magic; Sting subsequently became a host for his foster father's weakened soul, which allowed his Dragonification process to be halted due to the presence of new antibodies in his body. To ensure that Sting did not remember any of what had happened, Weisslogia implanted the memory that Sting had been forced to kill him and the results he would have actually achieved had such a thing been done.
aaron stewart's answer is more to the point as per the manga, according to the manga , all the dragon slayers are orphans and came back from past i,e; around 400 years with the help of lucy's mother who then opened the eclipse gate , that's how they come to the present and when they do i.e, the day all the dragons disappear and during the tartaros arc it's mentioned that the dragons hid them under the slayers so they could stop dragonification(sort of) and become something like acnologia .here both of them killing their dragons is just a memory implant maybe...