QM says that states which are simultaneous eigenstates of two commuting observables are allowed. If you don't have such states to start with you can construct them with the Gramm-Schmidt orthogonalization procedure. Consider the excited states of a nucleus. (They can be considered eigenstates of the Hamiltionian even though their lifetime is not infinite). Since the parity operator commutes with the Hamiltonian (ignore weak interactions), QM says simultaneous eigenstates of the Hamiltonian and of parity can be constructed. But tables of nuclear excited states determined experimentally are always labeled with their spin and parity. That is, experimentally measured energy eigenstates are also eigenstates of parity. Why? QM only says you can construct such states; it doesn't say that states occurring in nature must already be simultaneous eigenstates. It should be perfectly permissible to have nuclear excited states that are Hamiltonian eigenstates but are a linear combination of parity eigenstates. Why do we never see such states? If they do exist, but experimentalists like to assign quantum numbers, there should be in the tables degenerate states, one with positive parity, one with negative parity, with the same energy. One might think one could say the same thing about energy and angular momentum. Why are energy states always also eigenstates of angular momentum? But in this case, the Hamiltonian is a function of angular momentum, so states of different angular momentum have different energy and the question does not arise. That is not the case with parity.