Languages whose Parikh image is recognizable
I seems that I was quite confused back then. This language class is precisely the language class of those languages, whose commutative closure is regular.
Let $\Sigma = \{a_1, \ldots, a_k\}$. One implication is given in the question. For the other, suppose the commutative closure of $L \subseteq \Sigma^*$ is regular, i.e., the set $p^{-1}(p(L))$ is regular.
Then,  $p^{-1}(p(L)) \cap a_1^* \cdots a_k^*$ is regular, bounded and has the same Parikh image as $L$ or its commutative closure. By an old result of Ginsburg (see below), we can write this language as a finite union of languages of the form $A_1 \cdots A_k$, where each $A_i$ is a regular subset of $a_i^*$. Now, the Parikh image of one such set has the form $R_1 \times \ldots \times R_k$ with $R_i \subseteq \mathbb N$ recognizable. So, by a condition written in the question, the Parikh image of $L$ is recognizable.
Alternatively, without citing other results, suppose we have a morphism $h : \Sigma^* \to M$ with $M$ finite such that $p^{-1}(p(L)) = h^{-1}(h(L))$. We can assume $M$ is commutative (for example, by taking the syntactic monoid).
Hence, the map $\varphi : \mathbb N_0^k \to M$ given by $\varphi(n_1, \ldots, n_k) = h(a_1)^{n_1} \cdots h(a_k)^{n_k}$ is a morphism.
Then, $p(L) = \varphi^{-1}(h(L))$ and so $p(L)$ is recognizable.
Now, identifying language classes whose commutative closure is regular is quite an active field of research. For example, see the second reference below.
S. Ginsburg, Bounded Regular Sets, Amer. Math. Soc. 17 (1966), 1043-1049.
Antonio Cano Gómez, Giovanna Guaiana, Jean-Éric Pin:
Regular languages and partial commutations. Inf. Comput. 230: 76-96 (2013)