This is an edit of the prior question.
Is it consistent with $\sf ZF$ to have a family of pairwise disjoint sets of sets of naturals, that is larger in cardinality than the set of reals. That is: $$\exists X: |X|> |\mathbb R| \land \forall a,b \in X \, (a \subseteq \mathcal P (\omega) \land a \cap b =\emptyset)$$
Where cardinality $``||"$ is defined after Scott's.
This clearly violates the partition principle, so I think its consistent with $\sf ZF$
If the above is consistent, then call any set that has that quality as "Large partition", so the above statement would be saying that there exists a set $X$ that is a large partition on $\mathcal P(\omega)$, abbreviated formually as: $$ \exists X: X \ Lp \ \mathcal P(\omega)$$
Now do we have the following as a theorem of $\sf ZF$:
$$\exists X: X \ Lp \ \mathcal P(\omega) \to \exists Y: Y \ Lp \ \mathcal P(\omega) \land \forall p \in Y: \neg od(p)$$
Where $``od"$ stands for ordinal definable, defined as:
$$od(p) \iff \exists \varphi \exists \theta \exists a < \theta : p=\{y \in V_\theta \mid V_\theta \models \varphi(y,\alpha)\}$$
So this theorem is saying: if there is a large partition on the sets of naturals, then there is a large partition on them into non ordinal definable pieces.
Possibly even more daunting is the opposite question, that is all the pieces being ordinal definable, that is:
$$\exists X: X \ Lp \ \mathcal P(\omega) \to \exists Y: Y \ Lp \ \mathcal P(\omega) \land \forall p \in Y: od(p)$$
Yes, that's possible.
Start with $L$, but really any model of $\sf CH$ would work, and actually any model of $\sf ZFC$ would work. It's just to simplify things.
Adding an $\omega_2\times\omega$ Cohen reals, and consider permutations that move each $\omega$-block separately, and consider finite or even countable supports, in fact even $\aleph_1$-sized supports do the work here. (By which I mean you freeze the entire block, but only finitely/countably/$\aleph_1$ of them at a time)
This defines a model of $\sf ZF$ (with Dependent Choice if you allow countable support, and $\sf DC_{\omega_1}$ if you allow $\aleph_1$-sized supports) in which there is a surjection onto $\omega_2$, but there is no injection from $\omega_2$ into the reals.
However, this surjection itself, and the partition it defines, is not ordinal definable, since it is generic to a homogeneous forcing. So in particular none of its parts are ordinal definable.
Now, if you want something better, you can look, in the same model, at the surjection from $\Bbb R^\omega$ onto $[\Bbb R]^\omega$ mapping each sequence of reals to its range (adding the natural numbers if the range is finite). Crucially, this will also not admit an inverse, because there is an injection from $\omega_2$ into $[\Bbb R]^\omega$.
The partition of $\Bbb R$ given by $[\Bbb R]^\omega$ is ordinal definable, since I literally gave you the definition. But consider each Cohen real as a sequence of reals in its own right, then the fact we can switch the particular index of each Cohen real within a given $\omega$-block means that there cannot be an ordinal definition of this range, since it would have to be satisfied by many other Cohen reals as well.