  In this paper, we introduce the notions of $p$-Hermitian-symplectic and
$p$-pluriclosed compact complex manifolds as generalisations for an arbitrary
positive integer $p$ not exceeding the complex dimension of the manifold of the
standard notions of Hermitian-symplectic and SKT manifolds that correspond to
the case $p=1$. We then notice that these two properties are equivalent on
$\partial\bar{\partial}$-manifolds and go on to prove that in (smooth) complex
analytic families of $\partial\bar{\partial}$-manifolds, they are deformation
open. Concerning closedness results, we prove that the cones $\mathcal{A}_p$,
resp. $\mathcal{C}_p$, of Aeppli cohomology classes of strictly weakly positive
$(p,p)$-forms $\Omega$ that are $p$-pluriclosed, resp.
$p$-Hermitian-symplectic, must be equal on the limit fibre if they are equal on
the other fibres and if some rather weak $\partial\bar{\partial}$-type
assumptions are made on the other fibres.
