I must be out of it, because I thought that there had already been a definitive measurement of such spin-triplet superconductivity before this. But based on this latest report, it appears that NMR measurement of such superconductivity has not been "definitive" until now.
To clarify, the Cooper pair states in most superconducting material are formed due to the pairing of electrons forming what is known as the singlet state. This is when the spin of one electron pairs with another electron that has an opposite direction of spin. Via Pauli Exclusion principle (or rules for fermions), these two electrons can be quite close to one another in space because they do not occupy the same spin states.
However, the physics also allows for the two electrons to pair up with their spins in the same direction. Unfortunately, again due to Pauli, because they have the same spin state, their spatial state cannot be the same, i.e. they can't be too close to one another. This is a more difficult way to sustain this pair. The reason why this is called the triplet state, as opposed to the single state of the more common pairing, is because quantum mechanics allows for three possible ways for these electrons to pair up (see figure below) when we make a measurement along a particular direction.