No. Kids deal with all kinds of different situations where they need to learn to navigate through other people's bias, not just the CDing.
Their daughter will develop her own sensitivity as to when it is and it is not appropriate to share the information, and with whom. Much of it will depend on her own resilience of spirit. She may grow up to be militant in her support of LGBT rights and defy anyone who dares to make any comments to her face. Or, she may have a more quiet personality and will prefer to keep things private and only tell trusted friends. There are other factors that will influence how she handles it, namely how liberal her friends and her school environment are.
Her strongest asset is having *both* her parents present a united front as to how they wish to handle the situation.