Admitting into a nursing home; do I also need a POA?

Need a POA for what? What role are you wanting to play?

There are two kinds of POAs. One is for making medical decisions, and decisions about the person, such as where he lives. Apparently this is your mother for your dad.

The other is financial. The financial POA can pay bills, handle the bank account, etc. Does anyone have this kind of POA for your dad?

It should be noted that the POA acts on behalf of the principal. As long as your dad is not incompetent he can say what happens to his money and where he will live.

In general, it is not a good idea for two people to be equals on either POA. Consider this:
you and your mother each have medical POA authority
your dad is unconscious and unable to speak for himself
the doctor asks whether procedure A should be tried for your dad
Mom says NO. You say YES.
Now what? How is the doctor supposed to know whether to go ahead or not?

One person should be financial POA and one person should be medical POA. (The same person can be both.) There can (and should) be a backup person assigned. This doesn't come into play unless the primary POA is not able or willing to make the decision.