I don't know if the parity operator commute with the isospin operator T for any representation, i.e., [P,T_i]=0 ? i=1,2,3.
If two operators don't commute, an eigenstate of one operator cannot be an eigenstate of the other. The famous example here is that a system with definite angular momentum along one axis, represented by an eigenfunction of $J_z$, isn't at the same time an eigenstate of $J_x$ or $J_y$.
Since there are nuclear states where parity and total isospin are both good quantum numbers, those operators must commute.
If the eigenstates of one projection of the isospin operator (usually $T_3$) are the proton and neutron, the parity operator must commute with that projection as well. We don't live in a universe where a nuclear state can have an indeterminate electric charge, so all nuclear states are eigenstates of $T_3$ and no one cares about $T_1$ and $T_2$ by themselves. But if you combine them to make raising and lowering operators, you find that those don't modify parity either.