Not too sure if this is true, but I heard balloon parrots are actually just pink convicts. Generally speaking, male parrot cichlids are born sterile as they are hybrids. They may lay eggs and spawn with another fish but those eggs very rarely hatch and it is even more rare for any fry to survive very long.
If you look at this link and read the description for the fish it states they "breed easily". Without injecting males with hormones to increase fertility this is not true of actual blood parrots. Convicts on the other hand breed like rabbits. Apparently the link provided offers one of the few places to buy these fish.
I try to stay away from the "balloon" varieties as while their bodies have been shortened their organs remain the same size as a normal fish of their species. With their compacted bodies (bend spines usually) and their normal sized organs they are more prone to health issues, just like humans effected with dwarfism. Many of these fish die before reaching maturity and those that survive often have difficulty swimming and eating.