How do the family dynamics and child-rearing work here? I'm intrigued.
In my campaign, there are a few dragon societies, and a dragon that is a part of such a society often has dragonborn family or allies that help to rear the young. Lone dragons have to juggle it all themselves (unless they have a partner), and can get annoyed if they can't get their beauty sleep. Dragon culture is matriarchal, as the mother is the source of life. No matter the nature of the fertilizing father's color, the mother always gives birth to children matching her coloration. There are no mottled red/white or "pink" hybrids, for instance.
Dragonborn from dragons are gestated longer, and hatch more maturely grown than humanoid infants, their eggs and size being comparable to the dragon hatchling's. They also tend to hatch before their dragon sibling, being considered "older" and are often encouraged to become protectors of their younger sibling. After hatching, they develop similarly along age lines, but the dragon will ultimately physically eclipse their dragonborn siblings in size and power.
Good-aligned dragons usually treat dragonborn as kin, whether children, sibling, or cousin, though everyone knows that the dragon is more valued due to its rarity. While this can lead to jealousy, good dragons usually try to teach their children noble qualities and empathy and give them attention and purpose beyond comparing themselves to their sibling.
Evil-aligned dragons can often provide a toxic, abusive family environment, but dragonborn can be useful spawn, and they can be brainwashed to be very loyal and proud servants in service to evil. Sometimes chromatic dragonborn of evil dragons are sent away to broodgrounds ("military schools") or are sold by untrusting parents, because their learned ego and jealousy can be a danger to the real dragon egg.
Good and Evil are only tendencies for the respective dragon breeds, not universal truths. They can make their own alignment choices, influenced by their culture, religion, and personality. Tiamat worshippers tend not to be benevolent parents.