Take a compact complex torus with a Kähler structure, say, an abelian variety, for example. Then there are lots of non-vanishing closed parallel $(1,1)$-forms, and their products will generate $(p,p)$-forms.
More generally, ifIf $(X,\omega)$ is a Hermitian symmetric space, there will be many parallel parallel $(p,p)$ forms-forms (and they will all be closed). Exactly how many depends on the symmetric space.
Also, ifIf $(X,\omega)$ is a product of two Kähler manifolds, there will be more $(p,p)$ forms than just the powers of $\omega$. For example, if $X = X_1\times X_2$ and $\omega = \pi_1^*\omega_1 + \pi_2^*\omega_2$, where $(X_i,\omega_i)$ are compact Kähler manifolds, and $\pi_i:X\to X_i$ is the projection, then $\pi_i^*\omega_i$ for $i=1,2$ are parallel $(1,1)$-forms and the sub-algebra they generate consists entirely of parallel $(p,p)$-forms.
Conversely, if $(X,\omega)$ is not locally a product or locally symmetric, then the only parallel $(p,p)$-forms are the constant multiples of $\omega^p$.
This follows from the classification of holonomy groups of irreducible manifolds, since, if $(X,\omega)$ is an irreducible Kähler complex $n$-manifold and is not locally symmetric, then, by Berger's Theorem, its holonomy group is isomorphic to one of $\mathrm{U}(n)$, $\mathrm{SU}(n)$, or $\mathrm{Sp}(\tfrac12n)$. In all of these cases, any parallel $(p,p)$-form is a constant multiple of $\omega^p$.