How to show that a function is not a PRP? So I have to answer these questions, but I honestly have no idea what I am doing or where to begin. Can I get some advice on how to tackle these?

Consider a PRP candidate function $f$ on $K = X = \{0, 1\}^n$ which is a permutation on $X$ for every key in $K$, but for all $k, x \in \{0, 1\}^n$, it satisfies $\operatorname{parity}(f_k(x)) = \operatorname{parity}(x)$.


Show that $f$ is not secure as a PRP.
Show that the CBC encryption mode with block cipher $f$ is not indistinguishable.
Show that the CTR encryption mode with block cipher $f$ is not indistinguishable.
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 $g$.
$$\Pr[\operatorname{parity}(g(x)) = \operatorname{parity}(x)]=??$$