Let $\mathfrak{p}\subseteq \mathfrak{gl}_n(\mathbb{C})$ be a parabolic algebra of a parabolic group $P\subseteq GL_n(\mathbb{C})$.
What is the difference among the nilradical of $\mathfrak{p}$, the nilpotent ideal of $\mathfrak{p}$, and the Lie algebra of the intersection of the kernels of all homomorphisms from $P$ to the multiplicative group in $\mathbb{C}$?
The nilradical of $\mathfrak{p}$ is the largest nilpotent ideal of $\mathfrak{p}$. The phrase "the nilpotent ideal of $\mathfrak{p}$" is ambiguous because in general there are many such ideals. However there is always a largest such ideal, called the nilradical.
The intersection of the kernels of all homomorphisms from $P$ to $\mathbb{C}^\times$ is a subgroup of $P$ which contains the derived subgroup $P'$. In particular it is not a Lie algebra, so it cannot be meaningfully compared to the other objects in your question.
Furthermore, the Lie algebra of this kernel is in general strictly larger than the nilradical of $\mathfrak{p}$. They coincide precisely when $P$ is a minimal parabolic.