Signing up creates a family for you. If you did that before your co-parent invited you, you now own an empty family and their invite can't simply move you into theirs.
Opening the invite link explains this and offers to close your own family so you can join theirs. You have to tick a box to confirm.
What gets closed
Only a family that is solo and unpaid can be closed this way — one parent, nothing anyone else relies on, and no subscription. Anything you added to it is permanently deleted.
If your own family has a second parent in it, or a paid subscription, the app refuses. That data isn't ours to delete on your say-so. Email support and we'll work out what you want to keep.
If joining fails for any reason, nothing changes — you stay in your own family exactly as you were.