Parity operator with other operators

I think the first two things that is transformation of position and momentum operators are defined from definition...because by parity transformation sign of the position coordinates changes and time coordinate remain unchanged...so accordingly we get change in sign for position and momentum...once you do that then angular momentum should remain unchanged since it's given by cross product of position and momentum..these relations are valid in classical and quantum mechanics both..but here you can,as @Lubos Motl already pointed out the above identities are operator identities..so it should be true for any state of the hilbert space...e.g. $$p^{-1}hat{x}ppsi=sum_{x'}p^{-1}hat{x}p|x'ranglepsi(x')=sum_{x'}p^{-1}hat{x}|-x'ranglepsi(x')=sum_{x'}p^{-1}-x'|-x'ranglepsi(x')=sum_{x'}(-hat{x})|x'ranglepsi(x')=(-hat{x})psi$$