How to show that a function is not a PRP?
To show that a family of functions is not a PRP, you have to either show that the functions are not permutations or that they do not behave pseudo-randomly. As it is already established that the functions are in fact permutation you need to show the latter.

For a family of permutations to be a PRP means that it is computationally infeasible to distinguish a random member of the family (i.e. the function with a randmly chosen key) from a truly random permutation with non-negligible probability.
So what you need to do is present an efficient distinguisher. I.e., describe an algorithm that can in fact decide whether it is given access to a member of the family or to a random permutation.
To do that, you should consider the property that the parity of $f_k(x)$ is always equal to the parity of $x$. So for any fixed $x$ $$\Pr[\operatorname{parity}(f_k(x)) = \operatorname{parity}(x)]=1.$$ Now consider what the same probability is for a truly random permutation $g$.
$$\Pr[\operatorname{parity}(g(x)) = \operatorname{parity}(x)]=??$$