Is a paradoxical partition of a set smaller than the power set of that set? When it is said that almost every model of $\sf ZF + \neg AC$ do have paradoxical partitioning $\sf PP$, that is: there exists a set $X$ and a partition $P$ of $X$ that is strictly larger than $X$. Is it also provable that: $$|P|<|\mathcal P(X)| \text{ ?}$$

Yes. Note that trivially $|P|\leq|\mathcal{P}(X)|$ since $P\subseteq\mathcal{P}(X)$. If $|P|=|\mathcal{P}(X)|$, then we can define a surjection $X\to\mathcal{P}(X)$ by mapping each element of $X$ to the element of $P$ that contains it and then composing with a bijection $P\to\mathcal{P}(X)$. This is impossible by Cantor's theorem.