Frog and salamander
Reference: Plethodontidae
The group that changes its breathing apparatus partway through life — and includes hundreds of species with no lungs at all.
Scored on every axis
Three breathing systems in one animal
A tadpole has gills. An adult frog has lungs. Both stages also exchange gas across the skin, and in cold water a submerged frog can meet its needs through skin alone for long periods.
Then the case that sounds like an error and is not: the lungless salamanders are the largest salamander family in the world, hundreds of species, and none of them has lungs. Gas exchange is entirely through skin and mouth lining. The Bornean flat-headed frog is in the same position.
The constraint is that skin only works as a lung while it stays wet and demand stays low, which confines these animals to cool, damp, oxygen-rich places. Air-breathing and lung-owning are separate facts, and this is the group that proves it.