The odds of catching a Shiny Pokémon in Pokémon Brilliant Diamond and Brilliant Pearl are pretty low - 1 / 4,096 to be exact. These same probabilities apply to the possibility that an egg produced through breeding will also hatch a shiny Pokémon. Fortunately, when it comes to breeding, there is a way to increase the odds of acquiring a Shiny: the Masuda method.
Named after Game Freak director Junichi Masuda, the Masuda method is pretty simple. First, get a Pokémon via an online trade from a version of the game in a language other than yours. For example, if you are playing in English, you will want to trade with someone who is playing in Japanese or any other language besides English. Once you have your foreign Pokémon, take it to the Pokémon Kindergarten and leave it with the Pokémon you wish to breed it with.
As stated earlier, the odds of a Shiny Pokémon coming out of an egg are 1 / 4,096, but using the Masuda method dramatically increases the chance of acquiring a Shiny Pokémon to 1/683. It's still a possibility, but when it comes to brilliant hunting, you'll want to take advantage of all the perks you can.
To avoid having to find a foreign version of every Pokémon you're trying to get a Shiny version of, we highly recommend trying to find a foreign Ditto. Since Ditto can mate with any Pokémon regardless of gender or species, having a foreign Ditto allows you to use the Masuda Method on any Pokémon you wish without the need for additional leg work.
As for how you get foreign Pokémon, you might end up being paired with someone playing in a different language while in the Global Union Room, but that's not guaranteed. Fortunately, Shiny hunting is taken very seriously by the online Pokémon community, and therefore there are several social channels dedicated to helping players obtain foreign Pokémon, most notably Dittos.