The parent who apologizes still gets to be the parent.

Apologizing to a child is often avoided out of a worry about authority, as though admitting a mistake would hand over the wheel. In practice it does the opposite. A child who sees an adult name a mistake and repair it learns that being wrong is survivable, that people who love you come back, and that rules apply to everyone in the house. Authority built on never being wrong is brittle and children test it early.

Say what you did, without a but attached and without asking them to comfort you. Then carry on being the parent: the bedtime still stands, the answer is still no.