Will admitting I was wrong undermine my authority as a parent?

No, and the fear has it backwards.

The worry is that owning an error spends standing you can't earn back. Edward Tronick, whose research maps how young children build a sense of safety, found that they don't build it on a parent who never slips. That parent doesn't exist, and a child raised to believe they do is in for a hard correction. Kids build security on the experience of a break getting mended. A parent who can say "I got that wrong" is showing, live, that a rupture is survivable. The parent who can't is teaching that being wrong is a catastrophe, which is a heavier thing to hand a child than any one mistake.

Two things. A kid who remembers that you owned something isn't the disaster the fear imagines. Usually it means the repair landed and they're using the same move.

And the thing that genuinely erodes a parent's authority isn't the occasional admitted mistake. It's the reverse: the parent who can never be wrong, who rewrites what happened, who makes a child doubt their own memory to protect the parent's record. That's what kids quietly stop trusting. Owning a mistake reads as accountability. Refusing to ever own one is the actual leak. If a specific admission does get thrown back in a heated moment, that's a conversation about fairness, not evidence you should have skipped the repair.

The repair has a shape, and it's shorter than the fear makes it feel. Walk through it step by step here: How to repair a rupture with your child.