I'm of course assuming you mean blood parrot cichlids and the like... cause the other kind are saltwater fish..
Cichlids are kind of aggressive (even the blood parrots fight now and then) and probably not a great choice as they'd trigger aggression in the bettas. They can also grow to 8 inches.. and do eat small fish, so I'd be worried about the bettas being nipped when it got bigger.
They're lovely fish, though. Once day I'd like to keep cichlids.