Why did Sting and Rogue kill their father dragons? 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.