Difference between parallel and antiparallel spins?

[says]

What is the difference between parallel and antiparallel spins for a pair of nucleons?

My understanding is that nucleons have a strong tendency to pair - proton with proton, neutron with neutron, proton with neutron. When they pair their spins either:

cancel (spins pair antiparallel) pairing of a spin-up and spin-down nucleon
add (spins pair parallel) pairing of two nucleons with both spin up or both spin down

Am I understanding this correctly? - I'm thinking I might be missing something in regards to the Pauli Exclusion Principle. How can two nucleons with the same spin state pair?

 

[mfb]

How can two nucleons with the same spin state pair?

They cannot.
You can have two nucleons in different (spatial) states with the same spin orientation, however.

 

[says]

So nucleons have a strong tendency to pair due to the nuclear force, which has a spin component. When nucleons pair their spins will either be:
(Parallel) - Both spin the same way
(antiparallel) - One nucleon spin up and one spin down

Is there an energy difference between the two as well?

 

[says]

A more important question - Why are nucleons with parallel spin more strongly bound to each other than those having anti-parallel spin.