Does a new power attorney supersede a previous power attorney?

Read the document thoroughly. It will say IF the others are revoked.

A new POA doesn't replace an old POA unless, as GardenArtist wrote, the other is revoked. There can be more than one POA, so unless the other is revoked, both parties can act on behalf of the grantor.

It should specify that it rescinds, revokes and/or renders null and void any previous powers of attorney.