This is not an answer. Just a few well known facts.

Each $P$-invariant subset is also invariant with respect to the Levi part $L$ of $P$ and hence it decomposes into irreducibles for $L$.

The representation $\mathfrak{g/p}$ is as a $\mathfrak{p}$-representation isomorphic (via the Killing form) to the nilradical of $\mathfrak{p}$. Now this nilradical is in fact isomorphic to $k$-graded Lie algebra $\bigoplus_{i=1}^k \mathfrak{g}_i$ that is generated (as a Lie algebra) by $\mathfrak{g}_1$. Lie brackets $\mathfrak{g}_i \otimes \mathfrak{g}_j \to \mathfrak{g}_{i+j}$ for $i,j\in \{1,\ldots k\}$ are $L$-equivariant. (See e.g. section 3.1.2 in "parabook".)

One always have $P$-invariant subspaces given by filtration components $\mathfrak{g}^j = \bigoplus_{i=j}^k \mathfrak{g}_i$.

Consider a $P$-invariant subspace $V$. If $\emptyset \neq (V\cap \mathfrak{g}_i) \neq \mathfrak{g}_i$, then one can perhaps use the generating property of $\mathfrak{g}_1$ and $L$-equivariance of the Lie bracket $\mathfrak{g}_1\otimes (V\cap \mathfrak{g}_i) \to \mathfrak{g}_{i+1}$ to restrict possible $L$-types occurring in $V$.

But I don't know how the $L$-decompositions of $\mathfrak{g}_i$ look like.

This is not an answer. Just a few well known facts.

Each $P$-invariant subset is also invariant with respect to the Levi part $L$ of $P$ and hence it decomposes into irreducibles for $L$.

The representation $\mathfrak{g/p}$ is as a $\mathfrak{p}$-representation isomorphic (via the Killing form) to the nilradical of $\mathfrak{p}$. Now this nilradical is in fact isomorphic to $k$-graded Lie algebra $\bigoplus_{i=1}^k \mathfrak{g}_i$ that is generated (as a Lie algebra) by $\mathfrak{g}_1$. Lie brackets $\mathfrak{g}_i \otimes \mathfrak{g}_j \to \mathfrak{g}_{i+j}$ for $i,j\in \{1,\ldots k\}$ are $L$-equivariant. (See e.g. section 3.1.2 in Parabolic Geometries I Background and General Theory by Čap and Slovák.)

One always have $P$-invariant subspaces given by filtration components $\mathfrak{g}^j = \bigoplus_{i=j}^k \mathfrak{g}_i$.

Consider a $P$-invariant subspace $V$. If $\emptyset \neq (V\cap \mathfrak{g}_i) \neq \mathfrak{g}_i$, then one can perhaps use the generating property of $\mathfrak{g}_1$ and $L$-equivariance of the Lie bracket $\mathfrak{g}_1\otimes (V\cap \mathfrak{g}_i) \to \mathfrak{g}_{i+1}$ to restrict possible $L$-types occurring in $V$.

But I don't know how the $L$-decompositions of $\mathfrak{g}_i$ look like.